Triple point: Difference between revisions

Revision as of 13:18, 30 October 2013

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

See the article on topological spaces for basic definitions and examples, and see the article on topology for a brief history and description of the subject area. See Naive set theory, Axiomatic set theory, and Function for definitions concerning sets and functions. The following articles may also be useful. These either contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The list of general topology topics and the list of examples in general topology will also be very helpful.

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.

Template:CompactTOC8

A

Absolutely closed: See H-closed

Accessible: See $T_{1}$ .

Accumulation point: See limit point.

Alexandrov topology: A space X has the Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset.^[1]

Almost discrete: A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.

Approach space: An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.

B

Baire space

This has two distinct common meanings:

A space is a Baire space if the intersection of any countable collection of dense open sets is dense; see Baire space.
Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see Baire space (set theory).

Base: A collection B of open sets is a base (or basis) for a topology $τ$ if every open set in $τ$ is a union of sets in $B$ . The topology $τ$ is the smallest topology on $X$ containing $B$ and is said to be generated by $B$ .

Basis: See Base.

Borel algebra: The Borel algebra on a topological space $(X, τ)$ is the smallest $σ$ -algebra containing all the open sets. It is obtained by taking intersection of all $σ$ -algebras on $X$ containing $τ$ .

Borel set: A Borel set is an element of a Borel algebra.

Boundary: The boundary (or frontier) of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. Boundary of a set $A$ is denoted by $\partial A$ or $b d$ $A$ .

Bounded: A set in a metric space is bounded if it has finite diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A function taking values in a metric space is bounded if its image is a bounded set.

C

Category of topological spaces: The category Top has topological spaces as objects and continuous maps as morphisms.

Cauchy sequence: A sequence {x_n} in a metric space (M, d) is a Cauchy sequence if, for every positive real number r, there is an integer N such that for all integers m, n > N, we have d(x_m, x_n) < r.

Clopen set: A set is clopen if it is both open and closed.

Closed ball: If (M, d) is a metric space, a closed ball is a set of the form D(x; r) := {y in M : d(x, y) ≤ r}, where x is in M and r is a positive real number, the radius of the ball. A closed ball of radius r is a closed r-ball. Every closed ball is a closed set in the topology induced on M by d. Note that the closed ball D(x; r) might not be equal to the closure of the open ball B(x; r).

Closed set: A set is closed if its complement is a member of the topology.

Closed function: A function from one space to another is closed if the image of every closed set is closed.

Closure: The closure of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it. An element of the closure of a set S is a point of closure of S.

Closure operator: See Kuratowski closure axioms.

Coarser topology: If X is a set, and if T₁ and T₂ are topologies on X, then T₁ is coarser (or smaller, weaker) than T₂ if T₁ is contained in T₂. Beware, some authors, especially analysts, use the term stronger.

Comeagre: A subset A of a space X is comeagre (comeager) if its complement X\A is meagre. Also called residual.

Compact: A space is compact if every open cover has a finite subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact Hausdorff space is normal. See also quasicompact.

Compact-open topology: The compact-open topology on the set C(X, Y) of all continuous maps between two spaces X and Y is defined as follows: given a compact subset K of X and an open subset U of Y, let V(K, U) denote the set of all maps f in C(X, Y) such that f(K) is contained in U. Then the collection of all such V(K, U) is a subbase for the compact-open topology.

Complete: A metric space is complete if every Cauchy sequence converges.

Completely metrizable/completely metrisable: See complete space.

Completely normal: A space is completely normal if any two separated sets have disjoint neighbourhoods.

Completely normal Hausdorff: A completely normal Hausdorff space (or T₅ space) is a completely normal T₁ space. (A completely normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Every completely normal Hausdorff space is normal Hausdorff.

Completely regular: A space is completely regular if, whenever C is a closed set and x is a point not in C, then C and {x} are functionally separated.

Completely T₃: See Tychonoff.

Component: See Connected component/Path-connected component.

Connected: A space is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.

Connected component: A connected component of a space is a maximal nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a partition of that space.

Continuous: A function from one space to another is continuous if the preimage of every open set is open.

Continuum: A space is called a continuum if it a compact, connected Hausdorff space.

Contractible: A space X is contractible if the identity map on X is homotopic to a constant map. Every contractible space is simply connected.

Coproduct topology: If {X_i} is a collection of spaces and X is the (set-theoretic) disjoint union of {X_i}, then the coproduct topology (or disjoint union topology, topological sum of the X_i) on X is the finest topology for which all the injection maps are continuous.

Countable chain condition: A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.

Countably compact: A space is countably compact if every countable open cover has a finite subcover. Every countably compact space is pseudocompact and weakly countably compact.

Countably locally finite: A collection of subsets of a space X is countably locally finite (or σ-locally finite) if it is the union of a countable collection of locally finite collections of subsets of X.

Cover: A collection of subsets of a space is a cover (or covering) of that space if the union of the collection is the whole space.

Covering: See Cover.

Cut point: If X is a connected space with more than one point, then a point x of X is a cut point if the subspace X − {x} is disconnected.

D

Dense set: A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space.

Dense-in-itself set: A set is dense-in-itself if it has no isolated point.

Derived set: If X is a space and S is a subset of X, the derived set of S in X is the set of limit points of S in X.

Developable space: A toplogical space with a development.^[2]

Development: A countable collection of open covers of a toplogical space, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is disjoint from C.^[2]

Diameter: If (M, d) is a metric space and S is a subset of M, the diameter of S is the supremum of the distances d(x, y), where x and y range over S.

Discrete metric: The discrete metric on a set X is the function d : X × X → R such that for all x, y in X, d(x, x) = 0 and d(x, y) = 1 if x ≠ y. The discrete metric induces the discrete topology on X.

Discrete space: A space X is discrete if every subset of X is open. We say that X carries the discrete topology.^[3]

Discrete topology: See discrete space.

Disjoint union topology: See Coproduct topology.

Dispersion point: If X is a connected space with more than one point, then a point x of X is a dispersion point if the subspace X − {x} is hereditarily disconnected (its only connected components are the one-point sets).

Distance: See metric space.

Dunce hat (topology)

E

Entourage: See Uniform space.

Exterior: The exterior of a set is the interior of its complement.

F

F_σ set: An F_σ set is a countable union of closed sets.^[4]

Filter

A filter on a space X is a nonempty family F of subsets of X such that the following conditions hold:

The empty set is not in F.
The intersection of any finite number of elements of F is again in F.
If A is in F and if B contains A, then B is in F.

Finer topology: If X is a set, and if T₁ and T₂ are topologies on X, then T₂ is finer (or larger, stronger) than T₁ if T₂ contains T₁. Beware, some authors, especially analysts, use the term weaker.

Finitely generated: See Alexandrov topology.

First category: See Meagre.

First-countable: A space is first-countable if every point has a countable local base.

Fréchet: See T₁.

Frontier: See Boundary.

Full set: A compact subset K of the complex plane is called full if its complement is connected. For example, the closed unit disk is full, while the unit circle is not.

Functionally separated: Two sets A and B in a space X are functionally separated if there is a continuous map f: X → [0, 1] such that f(A) = 0 and f(B) = 1.

G

G_δ set: A G_δ set or inner limiting set is a countable intersection of open sets.^[4]

G_δ space: A space in which every closed set is a G_δ set.^[4]

Generic point: A generic point for a closed set is a point for which the closed set is the closure of the singleton set containing that point.^[5]

H

Hausdorff: A Hausdorff space (or T₂ space) is one in which every two distinct points have disjoint neighbourhoods. Every Hausdorff space is T₁.
H-closed: A space is H-closed, or Hausdorff closed or absolutely closed, if it is closed in every Hausdorff space containing it.
Hereditarily P: A space is hereditarily P for some property P if every subspace is also P.
Hereditary: A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.^[6] For example, second-countability is a hereditary property.
Homeomorphism: If X and Y are spaces, a homeomorphism from X to Y is a bijective function f : X → Y such that f and f⁻¹ are continuous. The spaces X and Y are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical.
Homogeneous: A space X is homogeneous if, for every x and y in X, there is a homeomorphism f : X → X such that f(x) = y. Intuitively, the space looks the same at every point. Every topological group is homogeneous.
Homotopic maps: Two continuous maps f, g : X → Y are homotopic (in Y) if there is a continuous map H : X × [0, 1] → Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X. Here, X × [0, 1] is given the product topology. The function H is called a homotopy (in Y) between f and g.
Homotopy: See Homotopic maps.
Hyper-connected: A space is hyper-connected if no two non-empty open sets are disjoint^[7] Every hyper-connected space is connected.^[7]

I

Identification map: See Quotient map.
Identification space: See Quotient space.
Indiscrete space: See Trivial topology.
Infinite-dimensional topology: See Hilbert manifold and Q-manifolds, i.e. (generalized) manifolds modelled on the Hilbert space and on the Hilbert cube respectively.
Inner limiting set: A G_δ set.^[4]
Interior: The interior of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set S is an interior point of S.
Interior point: See Interior.
Isolated point: A point x is an isolated point if the singleton {x} is open. More generally, if S is a subset of a space X, and if x is a point of S, then x is an isolated point of S if {x} is open in the subspace topology on S.
Isometric isomorphism: If M₁ and M₂ are metric spaces, an isometric isomorphism from M₁ to M₂ is a bijective isometry f : M₁ → M₂. The metric spaces are then said to be isometrically isomorphic. From the standpoint of metric space theory, isometrically isomorphic spaces are identical.
Isometry: If (M₁, d₁) and (M₂, d₂) are metric spaces, an isometry from M₁ to M₂ is a function f : M₁ → M₂ such that d₂(f(x), f(y)) = d₁(x, y) for all x, y in M₁. Every isometry is injective, although not every isometry is surjective.

K

Kolmogorov axiom: See T₀.

Kuratowski closure axioms

The Kuratowski closure axioms is a set of axioms satisfied by the function which takes each subset of X to its closure:

Isotonicity: Every set is contained in its closure.
Idempotence: The closure of the closure of a set is equal to the closure of that set.
Preservation of binary unions: The closure of the union of two sets is the union of their closures.
Preservation of nullary unions: The closure of the empty set is empty.

If c is a function from the power set of X to itself, then c is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on X by declaring the closed sets to be the fixed points of this operator, i.e. a set A is closed if and only if c(A) = A.

L

L-space: An L-space is a hereditarily Lindelöf space which is not hereditarily separable. A Suslin line would be an L-space.^[8]

Larger topology: See Finer topology.

Limit point: A point x in a space X is a limit point of a subset S if every open set containing x also contains a point of S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself.

Limit point compact: See Weakly countably compact.

Lindelöf: A space is Lindelöf if every open cover has a countable subcover.

Local base: A set B of neighbourhoods of a point x of a space X is a local base (or local basis, neighbourhood base, neighbourhood basis) at x if every neighbourhood of x contains some member of B.

Local basis: See Local base.

Locally closed subset: A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure.

Locally compact: A space is locally compact if every point has a local base consisting of compact neighbourhoods. Every locally compact Hausdorff space is Tychonoff.

Locally connected: A space is locally connected if every point has a local base consisting of connected neighbourhoods.

Locally finite: A collection of subsets of a space is locally finite if every point has a neighbourhood which has nonempty intersection with only finitely many of the subsets. See also countably locally finite, point finite.

Locally metrizable/Locally metrisable: A space is locally metrizable if every point has a metrizable neighbourhood.

Locally path-connected: A space is locally path-connected if every point has a local base consisting of path-connected neighbourhoods. A locally path-connected space is connected if and only if it is path-connected.

Locally simply connected: A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods.

Loop: If x is a point in a space X, a loop at x in X (or a loop in X with basepoint x) is a path f in X, such that f(0) = f(1) = x. Equivalently, a loop in X is a continuous map from the unit circle S¹ into X.

M

Meagre: If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable union of nowhere dense sets. If A is not meagre in X, A is of second category in X.^[9]

Metric: See Metric space.

Metric invariant: A metric invariant is a property which is preserved under isometric isomorphism.

Metric map: If X and Y are metric spaces with metrics d_X and d_Y respectively, then a metric map is a function f from X to Y, such that for any points x and y in X, d_Y(f(x), f(y)) ≤ d_X(x, y). A metric map is strictly metric if the above inequality is strict for all x and y in X.

Metric space

A metric space (M, d) is a set M equipped with a function d : M × M → R satisfying the following axioms for all x, y, and z in M:

d(x, y) ≥ 0
d(x, x) = 0
if d(x, y) = 0 then x = y (identity of indiscernibles)
d(x, y) = d(y, x) (symmetry)
d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

The function d is a metric on M, and d(x, y) is the distance between x and y. The collection of all open balls of M is a base for a topology on M; this is the topology on M induced by d. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable.

Metrizable/Metrisable: A space is metrizable if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable.

Monolith: Every non-empty ultra-connected compact space X has a largest proper open subset; this subset is called a monolith.

Moore space: A Moore space is a developable regular Hausdorff space.^[2]

N

Neighbourhood/Neighborhood: A neighbourhood of a point x is a set containing an open set which in turn contains the point x. More generally, a neighbourhood of a set S is a set containing an open set which in turn contains the set S. A neighbourhood of a point x is thus a neighbourhood of the singleton set {x}. (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.)

Neighbourhood base/basis: See Local base.

Neighbourhood system for a point x: A neighbourhood system at a point x in a space is the collection of all neighbourhoods of x.

Net: A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (x_α), where α is an index variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers with the usual ordering.

Normal: A space is normal if any two disjoint closed sets have disjoint neighbourhoods.^[4] Every normal space admits a partition of unity.

Normal Hausdorff: A normal Hausdorff space (or T₄ space) is a normal T₁ space. (A normal space is Hausdorff if and only if it is T₁, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.

Nowhere dense: A nowhere dense set is a set whose closure has empty interior.

O

Open cover: An open cover is a cover consisting of open sets.^[2]

Open ball: If (M, d) is a metric space, an open ball is a set of the form B(x; r) := {y in M : d(x, y) < r}, where x is in M and r is a positive real number, the radius of the ball. An open ball of radius r is an open r-ball. Every open ball is an open set in the topology on M induced by d.
Open condition: See open property.
Open set: An open set is a member of the topology.
Open function: A function from one space to another is open if the image of every open set is open.
Open property: A property of points in a topological space is said to be "open" if those points which possess it form an open set. Such conditions often take a common form, and that form can be said to be an open condition; for example, in metric spaces, one defines an open ball as above, and says that "strict inequality is an open condition".

P

Paracompact: A space is paracompact if every open cover has a locally finite open refinement. Paracompact implies metacompact.^[10] Paracompact Hausdorff spaces are normal.^[11]

Partition of unity: A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.

Path: A path in a space X is a continuous map f from the closed unit interval [0, 1] into X. The point f(0) is the initial point of f; the point f(1) is the terminal point of f.^[7]

Path-connected: A space X is path-connected if, for every two points x, y in X, there is a path f from x to y, i.e., a path with initial point f(0) = x and terminal point f(1) = y. Every path-connected space is connected.^[7]

Path-connected component: A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a partition of that space, which is finer than the partition into connected components.^[7] The set of path-connected components of a space X is denoted π₀(X).

Perfectly normal: a normal space which is also a G_δ.^[4]

π-base: A collection B of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includes a set from B.^[12]

Point: A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point".

Point of closure: See Closure.

Polish: A space is Polish if it is separable and topologically complete, i.e. if it is homeomorphic to a separable and complete metric space.

P-point: A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersections.

Pre-compact: See Relatively compact.

Product topology: If {X_i} is a collection of spaces and X is the (set-theoretic) product of {X_i}, then the product topology on X is the coarsest topology for which all the projection maps are continuous.

Proper function/mapping: A continuous function f from a space X to a space Y is proper if f⁻¹(C) is a compact set in X for any compact subspace C of Y.

Proximity space: A proximity space (X, δ) is a set X equipped with a binary relation δ between subsets of X satisfying the following properties:

For all subsets A, B and C of X,

A δ B implies B δ A
A δ B implies A is non-empty
If A and B have non-empty intersection, then A δ B
A δ (B ∪ C) iff (A δ B or A δ C)
If, for all subsets E of X, we have (A δ E or B δ E), then we must have A δ (X − B)

Pseudocompact: A space is pseudocompact if every real-valued continuous function on the space is bounded.

Pseudometric: See Pseudometric space.

Pseudometric space: A pseudometric space (M, d) is a set M equipped with a function d : M × M → R satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. The function d is a pseudometric on M. Every metric is a pseudometric.

Punctured neighbourhood/Punctured neighborhood: A punctured neighbourhood of a point x is a neighbourhood of x, minus {x}. For instance, the interval (−1, 1) = {y : −1 < y < 1} is a neighbourhood of x = 0 in the real line, so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0.

Q

Quasicompact: See compact. Some authors define "compact" to include the Hausdorff separation axiom, and they use the term quasicompact to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French.

Quotient map: If X and Y are spaces, and if f is a surjection from X to Y, then f is a quotient map (or identification map) if, for every subset U of Y, U is open in Y if and only if f^-1(U) is open in X. In other words, Y has the f-strong topology. Equivalently, $f$ is a quotient map if and only if it is the transfinite composition of maps $X \to X / Z$ , where $Z \subset X$ is a subset. Note that this doesn't imply that f is an open function.

Quotient space: If X is a space, Y is a set, and f : X → Y is any surjective function, then the quotient topology on Y induced by f is the finest topology for which f is continuous. The space X is a quotient space or identification space. By definition, f is a quotient map. The most common example of this is to consider an equivalence relation on X, with Y the set of equivalence classes and f the natural projection map. This construction is dual to the construction of the subspace topology.

R

Refinement: A cover K is a refinement of a cover L if every member of K is a subset of some member of L.
Regular: A space is regular if, whenever C is a closed set and x is a point not in C, then C and x have disjoint neighbourhoods.
Regular Hausdorff: A space is regular Hausdorff (or T₃) if it is a regular T₀ space. (A regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.)
Regular open: An open subset U of a space X is regular open if it equals the interior of its closure; simiarly, a regular closed set is equal to the closure of is interior.^[13] An example of a non-regular open set is the set U = Template:Open-open ∪ Template:Open-open in R with its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsets of a space form a complete Boolean algebra.^[13]
Relatively compact: A subset Y of a space X is relatively compact in X if the closure of Y in X is compact.
Residual: If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X. Also called comeagre or comeager.
Resolvable: A topological space is called resolvable if it is expressible as the union of two disjoint dense subsets.
Rim-compact: A space is rim-compact if it has a base of open sets whose boundaries are compact.

S

S-space: An S-space is a hereditarily separable space which is not hereditarily Lindelöf.^[8]

Scattered: A space X is scattered if every nonempty subset A of X contains a point isolated in A.

Scott: The Scott topology on a poset is that in which the open sets are those Upper sets inaccessible by directed joins.^[14]

Second category: See Meagre.

Second-countable: A space is second-countable or perfectly separable if it has a countable base for its topology.^[4] Every second-countable space is first-countable, separable, and Lindelöf.

Semilocally simply connected: A space X is semilocally simply connected if, for every point x in X, there is a neighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simply connected space and every locally simply connected space is semilocally simply connected. (Compare with locally simply connected; here, the homotopy is allowed to live in X, whereas in the definition of locally simply connected, the homotopy must live in U.)

Semiregular: A space is semiregular if the regular open sets form a base.

Separable: A space is separable if it has a countable dense subset.^[4]^[9]

Separated: Two sets A and B are separated if each is disjoint from the other's closure.

Sequentially compact: A space is sequentially compact if every sequence has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.

Short map: See metric map

Simply connected: A space is simply connected if it is path-connected and every loop is homotopic to a constant map.

Smaller topology: See Coarser topology.

Sober: In a sober space, every irreducible closed subset is the closure of exactly one point: that is, has a unique generic point.^[15]

Star: The star of a point in a given cover of a topological space is the union of all the sets in the cover that contain the point. See star refinement.

$f$ -Strong topology: Let $f : X \to Y$ be a map of topological spaces. We say that $Y$ has the $f$ -strong topology if, for every subset $U \subset Y$ , one has that $U$ is open in $Y$ if and only if $f^{- 1} (U)$ is open in $X$

Stronger topology: See Finer topology. Beware, some authors, especially analysts, use the term weaker topology.

Subbase: A collection of open sets is a subbase (or subbasis) for a topology if every non-empty proper open set in the topology is a union of finite intersections of sets in the subbase. If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B; this topology consists of the empty set, X and all unions of finite intersections of elements of B.

Subbasis: See Subbase.

Subcover: A cover K is a subcover (or subcovering) of a cover L if every member of K is a member of L.

Subcovering: See Subcover.
Submaximal space: A topological space is said to be submaximal if every subset of it is locally closed, that is, every subset is the intersection of an open set and a closed set.

Here are some facts about submaximality as a property of topological spaces:

Every door space is submaximal.
Every submaximal space is weakly submaximal viz every finite set is locally closed.
Every submaximal space is irresolvable^[16]

Subspace: If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced by T consists of all intersections of open sets in T with A. This construction is dual to the construction of the quotient topology.

T

T₀: A space is T₀ (or Kolmogorov) if for every pair of distinct points x and y in the space, either there is an open set containing x but not y, or there is an open set containing y but not x.

T₁: A space is T₁ (or Fréchet or accessible) if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T₀; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T₁ if all its singletons are closed. Every T₁ space is T₀.

T₂: See Hausdorff space.

T₃: See Regular Hausdorff.

T_3½: See Tychonoff space.

T₄: See Normal Hausdorff.

T₅: See Completely normal Hausdorff.

Top: See Category of topological spaces.

Topological invariant: A topological invariant is a property which is preserved under homeomorphism. For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces.

Topological space: A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms:

The empty set and X are in T.
The union of any collection of sets in T is also in T.
The intersection of any pair of sets in T is also in T.

The collection T is a topology on X.

Topological sum: See Coproduct topology.

Topologically complete: A space is topologically complete if it is homeomorphic to a complete metric space.

Topology: See Topological space.

Totally bounded: A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by open balls of radius r. A metric space is compact if and only if it is complete and totally bounded.

Totally disconnected: A space is totally disconnected if it has no connected subset with more than one point.

Trivial topology: The trivial topology (or indiscrete topology) on a set X consists of precisely the empty set and the entire space X.

Tychonoff: A Tychonoff space (or completely regular Hausdorff space, completely T₃ space, T_3.5 space) is a completely regular T₀ space. (A completely regular space is Hausdorff if and only if it is T₀, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.

U

Ultra-connected: A space is ultra-connected if no two non-empty closed sets are disjoint.^[7] Every ultra-connected space is path-connected.

Ultrametric: A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: for all x, y, z in M, d(x, z) ≤ max(d(x, y), d(y, z)).

Uniform isomorphism: If X and Y are uniform spaces, a uniform isomorphism from X to Y is a bijective function f : X → Y such that f and f⁻¹ are uniformly continuous. The spaces are then said to be uniformly isomorphic and share the same uniform properties.

Uniformizable/Uniformisable: A space is uniformizable if it is homeomorphic to a uniform space.

Uniform space: A uniform space is a set U equipped with a nonempty collection Φ of subsets of the Cartesian product X × X satisfying the following axioms:

if U is in Φ, then U contains { (x, x) | x in X }.
if U is in Φ, then { (y, x) | (x, y) in U } is also in Φ
if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
if U and V are in Φ, then U ∩ V is in Φ
if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U.

The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.

Uniform structure: See Uniform space.

W

Weak topology: The weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
Weaker topology: See Coarser topology. Beware, some authors, especially analysts, use the term stronger topology.
Weakly countably compact: A space is weakly countably compact (or limit point compact) if every infinite subset has a limit point.
Weakly hereditary: A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.
Weight: The weight of a space X is the smallest cardinal number κ such that X has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is well-ordered.)
Well-connected: See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)

Z

Zero-dimensional: A space is zero-dimensional if it has a base of clopen sets.^[17]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Also available as Dover reprint.

External links

A glossary of definitions in topology

↑ Vickers (1989) p.22
↑ ^2.0 ^2.1 ^2.2 ^2.3 Steen & Seebach (1978) p.163
↑ Steen & Seebach (1978) p.41
↑ ^4.0 ^4.1 ^4.2 ^4.3 ^4.4 ^4.5 ^4.6 ^4.7 Steen & Seebach (1978) p.162
↑ Vickers (1989) p.65
↑ Steen & Seebach p.4
↑ ^7.0 ^7.1 ^7.2 ^7.3 ^7.4 ^7.5 Steen & Seebach (1978) p.29
↑ ^8.0 ^8.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^9.0 ^9.1 Steen & Seebach (1978) p.7
↑ Steen & Seebach (1978) p.23
↑ Steen & Seebach (1978) p.25
↑ Hart, Nagata, Vaughan Sect. d-22, page 227
↑ ^13.0 ^13.1 Steen & Seebach (1978) p.6
↑ Vickers (1989) p.95
↑ Vickers (1989) p.66
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Steen & Seebach (1978) p.33

[1] Vickers (1989) p.22

[ss163-2] 2.0 ^2.1 ^2.2 ^2.3 Steen & Seebach (1978) p.163

[ss41-3] Steen & Seebach (1978) p.41

[ss162-4] 4.0 ^4.1 ^4.2 ^4.3 ^4.4 ^4.5 ^4.6 ^4.7 Steen & Seebach (1978) p.162

[5] Vickers (1989) p.65

[ss4-6] Steen & Seebach p.4

[ss29-7] 7.0 ^7.1 ^7.2 ^7.3 ^7.4 ^7.5 Steen & Seebach (1978) p.29

[GKW290-8] 8.0 ^8.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[ss7-9] 9.0 ^9.1 Steen & Seebach (1978) p.7

[ss23-10] Steen & Seebach (1978) p.23

[ss25-11] Steen & Seebach (1978) p.25

[12] Hart, Nagata, Vaughan Sect. d-22, page 227

[Steen_&_Seebach_1978_p.6-13] 13.0 ^13.1 Steen & Seebach (1978) p.6

[14] Vickers (1989) p.95

[15] Vickers (1989) p.66

[16] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[ss33-17] Steen & Seebach (1978) p.33

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

@@ Line 1: / Line 1: @@
-Hello, I'm Tiara, a 28 year old from Zabrze, Poland.<br>My hobbies include (but are not limited to) Home Movies, Rock climbing and watching How I Met Your Mother.<br><br>my webpage :: [http://Batwiki.Svenskabatmarknaden.se/wiki/Anv%C3%A4ndare:Jeffrey44V FIFA 15 Coin Hack]
+This is a glossary of some terms used in the branch of [[mathematics]] known as [[topology]]. Although there is no absolute distinction between different areas of topology, the focus here is on [[general topology]]. The following definitions are also fundamental to [[algebraic topology]], [[differential topology]] and [[geometric topology]].
+See the article on [[topological space]]s for basic definitions and examples, and see the article on [[topology]] for a brief history and description of the subject area. See [[Naive set theory]], [[Axiomatic set theory]], and [[Function (mathematics)|Function]] for definitions concerning sets and functions. The following articles may also be useful.  These either contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The [[list of general topology topics]] and the [[list of examples in general topology]] will also be very helpful.
+* [[Compact space]]
+* [[Connected space]]
+* [[Continuity (topology)|Continuity]]
+* [[Metric space]]
+* [[Separated sets]]
+* [[Separation axiom]]
+* [[Topological space]]
+* [[Uniform space]]
+{{See also|Glossary of Riemannian and metric geometry}}
+All spaces in this glossary are assumed to be [[topological space]]s unless stated otherwise.
+__NOTOC__
+{{compactTOC8|side=yes|top=yes|num=yes}}
+== A ==
+;Absolutely closed: See ''H-closed''
+;Accessible: See '''[[T1 space|<math>T_1</math>]]'''.
+;Accumulation point: See [[limit point]].
+;[[Alexandrov topology]]:  A space ''X'' has the [[Alexandrov topology]] (or is '''finitely generated''') if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the [[upper set]]s of a [[poset]].<ref>Vickers (1989) p.22</ref>
+;Almost discrete: A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.
+;[[Approach space]]:  An [[approach space]] is a generalization of metric space based on point-to-set distances, instead of point-to-point.
+== B ==
+;Baire space: This has two distinct common meanings:
+:#A space is a '''Baire space''' if the intersection of any [[countable]] collection of dense open sets is dense; see [[Baire space]].
+:#'''Baire space''' is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see [[Baire space (set theory)]].
+;[[Base (topology)|Base]]: A collection ''B'' of open sets is a [[base (topology)|base]] (or '''basis''') for a topology <math>\tau</math> if every open set in <math>\tau</math> is a union of sets in <math> B </math>. The topology <math>\tau</math> is the smallest topology on <math>X</math> containing <math>B</math> and is said to be generated by <math>B</math>.
+;[[Basis (topology)|Basis]]: See '''[[Base (topology)|Base]]'''.
+;[[Borel algebra]]: The [[Borel algebra]] on a topological space <math> (X,\tau)</math> is the smallest [[Sigma-algebra|<math>\sigma</math>-algebra]] containing all the open sets. It is obtained by taking intersection of all <math>\sigma</math>-algebras on <math> X </math> containing <math> \tau </math>.
+;Borel set: A Borel set is an element of a Borel algebra.
+;[[Boundary (topology)|Boundary]]: The [[boundary (topology)|boundary]] (or '''frontier''') of a set is the set's closure minus its interior.  Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. Boundary of a set <math> A </math> is denoted by <math> \partial A</math> or <math>bd</math> <math>A</math>.
+;[[Bounded set|Bounded]]: A set in a metric space is [[bounded set|bounded]] if it has [[finite set|finite]] diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A [[function (mathematics)|function]] taking values in a metric space is [[bounded function|bounded]] if its [[image (functions)|image]] is a bounded set.
+== C ==
+;[[Category of topological spaces]]: The [[category theory|category]] '''[[Category of topological spaces|Top]]''' has [[topological space]]s as [[object (category theory)|objects]] and [[continuous map]]s as [[morphism]]s.
+;[[Cauchy sequence]]: A [[sequence]] {''x''<sub>''n''</sub>} in a metric space (''M'', ''d'') is a [[Cauchy sequence]] if, for every [[positive number|positive]] [[real number]] ''r'', there is an [[integer]] ''N'' such that for all integers ''m'', ''n'' > ''N'', we have ''d''(''x''<sub>''m''</sub>, ''x''<sub>''n''</sub>) < ''r''.
+;[[Clopen set]]: A set is [[clopen set|clopen]] if it is both open and closed.
+;Closed ball: If (''M'', ''d'') is a [[metric space]], a closed ball is a set of the form ''D''(''x''; ''r'') := {''y'' in ''M'' : ''d''(''x'', ''y'') ≤ ''r''}, where ''x'' is in ''M'' and ''r'' is a [[positive number|positive]] [[real number]], the '''radius''' of the ball. A closed ball of radius ''r'' is a '''closed ''r''-ball'''. Every closed ball is a closed set in the topology induced on ''M'' by ''d''. Note that the closed ball ''D''(''x''; ''r'') might not be equal to the [[closure (topology)|closure]] of the open ball ''B''(''x''; ''r'').
+;[[Closed set]]: A set is [[Closed set|closed]] if its complement is a member of the topology.
+;[[Closed function]]: A function from one space to another is closed if the [[image (mathematics)|image]] of every closed set is closed.
+;[[Closure (topology)|Closure]]: The [[closure (topology)|closure]] of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it. An element of the closure of a set ''S'' is a '''point of closure''' of ''S''.
+;Closure operator: See '''[[Kuratowski closure axioms]]'''.
+;[[Coarser topology]]: If ''X'' is a set, and if ''T''<sub>1</sub> and ''T''<sub>2</sub> are topologies on ''X'', then ''T''<sub>1</sub> is [[coarser topology|coarser]] (or '''smaller''', '''weaker''') than ''T''<sub>2</sub> if ''T''<sub>1</sub> is contained in ''T''<sub>2</sub>. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''stronger'''.
+;Comeagre: A subset ''A'' of a space ''X'' is '''comeagre''' ('''comeager''') if its [[complement (set theory)|complement]] ''X''\''A'' is [[meagre set|meagre]]. Also called '''residual'''.
+;[[Compact space|Compact]]: A space is [[compact space|compact]] if every open cover has a [[finite set|finite]] subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact [[Hausdorff space]] is normal. See also '''quasicompact'''.
+;[[Compact-open topology]]: The [[compact-open topology]] on the set ''C''(''X'', ''Y'') of all continuous maps between two spaces ''X'' and ''Y'' is defined as follows: given a compact subset ''K'' of ''X'' and an open subset ''U'' of ''Y'', let ''V''(''K'', ''U'') denote the set of all maps ''f'' in ''C''(''X'', ''Y'') such that ''f''(''K'') is contained in ''U''. Then the collection of all such ''V''(''K'', ''U'') is a subbase for the compact-open topology.
+;[[Complete space|Complete]]: A metric space is [[complete space|complete]] if every Cauchy sequence converges.
+;Completely metrizable/completely metrisable: See '''[[complete space]]'''.
+;Completely normal: A space is completely normal if any two separated sets have [[Disjoint sets|disjoint]] neighbourhoods.
+;Completely normal Hausdorff: A completely normal Hausdorff space (or [[T5 space|'''T<sub>5</sub>''' space]]) is a completely normal T<sub>1</sub> space. (A completely normal space is Hausdorff [[if and only if]] it is T<sub>1</sub>, so the terminology is [[consistent]].) Every completely normal Hausdorff space is normal Hausdorff.
+;[[Completely regular space|Completely regular]]: A space is [[Completely regular space|completely regular]] if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and {''x''} are functionally separated.
+;[[Completely T3 space|Completely T<sub>3</sub>]]: See '''[[Tychonoff space|Tychonoff]]'''.
+;Component: See '''[[Connected space|Connected component]]'''/'''Path-connected component'''.
+;[[Connected (topology)|Connected]]: A space is [[connected (topology)|connected]] if it is not the union of a pair of [[Disjoint sets|disjoint]] nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.
+;[[connected space|Connected component]]: A [[connected space|connected component]] of a space is a [[maximal set|maximal]] nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a [[partition of a set|partition]] of that space.
+;[[Continuity (topology)|Continuous]]: A function from one space to another is [[continuity (topology)|continuous]] if the [[preimage]] of every open set is open.
+;Continuum: A space is called a continuum if it a compact, connected Hausdorff space.
+;[[Contractible space|Contractible]]: A space ''X'' is contractible if the [[identity function|identity map]] on ''X'' is homotopic to a constant map. Every contractible space is simply connected.
+;Coproduct topology: If {''X''<sub>''i''</sub>} is a collection of spaces and ''X'' is the (set-theoretic) [[disjoint union]] of {''X''<sub>''i''</sub>}, then the coproduct topology (or '''disjoint union topology''', '''topological sum''' of the ''X''<sub>''i''</sub>) on ''X'' is the finest topology for which all the injection maps are continuous.
+;[[Countable chain condition]]: A space ''X'' satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.
+;Countably compact: A space is countably compact if every [[countable]] open cover has a [[finite set|finite]] subcover. Every countably compact space is pseudocompact and weakly countably compact.
+;Countably locally finite:  A collection of subsets of a space ''X'' is '''countably locally finite''' (or '''σ-locally finite''') if it is the union of a [[countable]] collection of locally finite collections of subsets of ''X''.
+;[[Cover (topology)|Cover]]: A collection of subsets of a space is a cover (or '''covering''') of that space if the union of the collection is the whole space.
+;Covering: See '''Cover'''.
+;Cut point: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a cut point if the subspace ''X'' − {''x''} is disconnected.
+== D ==
+;[[Dense set]]: A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space.
+;[[Dense-in-itself]] set: A set is dense-in-itself if it has no [[isolated point]].
+;Derived set: If ''X'' is a space and ''S'' is a subset of ''X'', the derived set of ''S'' in ''X'' is the set of limit points of ''S'' in ''X''.
+;Developable space: A toplogical space with a [[Development (topology)|development]].<ref name=ss163/>
+;[[Development (topology)|Development]]: A [[countable set|countable]] collection of [[open cover]]s of a toplogical space, such that for any closed set ''C'' and any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is [[disjoint sets|disjoint]] from ''C''.<ref name=ss163/>
+;Diameter: If (''M'', ''d'') is a metric space and ''S'' is a subset of ''M'', the diameter of ''S'' is the [[supremum]] of the distances ''d''(''x'', ''y''), where ''x'' and ''y'' range over ''S''.
+;Discrete metric: The discrete metric on a set ''X'' is the function ''d'' : ''X'' × ''X'' &nbsp;→&nbsp; '''[[real number|R]]''' such that for all ''x'', ''y'' in ''X'', ''d''(''x'', ''x'') = 0 and ''d''(''x'', ''y'') = 1 if ''x'' ≠ ''y''. The discrete metric induces the discrete topology on ''X''.
+;[[Discrete space]]: A space ''X'' is [[discrete space|discrete]] if every subset of ''X'' is open. We say that ''X'' carries the '''discrete topology'''.<ref name=ss41>Steen &amp; Seebach (1978) p.41</ref>
+;[[Discrete topology]]: See '''[[discrete space]]'''.
+;Disjoint union topology: See '''Coproduct topology'''.
+;[[Dispersion point]]: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a dispersion point if the subspace ''X'' − {''x''} is hereditarily disconnected (its only connected components are the one-point sets).
+;Distance: See '''[[metric space]]'''.
+;[[Dunce hat (topology)]]
+== E ==
+;[[Entourage (topology)|Entourage]]: See '''[[Uniform space]]'''.
+;Exterior: The exterior of a set is the interior of its complement.
+== F ==
+;[[F-sigma set|''F''<sub>σ</sub> set]]: An [[F-sigma set|''F''<sub>σ</sub> set]] is a [[countable]] union of closed sets.<ref name=ss162/>
+;[[Filter (mathematics)|Filter]]: A filter on a space ''X'' is a nonempty family ''F'' of subsets of ''X'' such that the following conditions hold:
+:# The [[empty set]] is not in ''F''.
+:# The intersection of any [[finite set|finite]] number of elements of ''F'' is again in ''F''.
+:# If ''A'' is in ''F'' and if ''B'' contains ''A'', then ''B'' is in ''F''.
+;[[Finer topology]]: If ''X'' is a set, and if ''T''<sub>1</sub> and ''T''<sub>2</sub> are topologies on ''X'', then ''T''<sub>2</sub> is [[finer topology|finer]] (or '''larger''', '''stronger''') than ''T''<sub>1</sub> if ''T''<sub>2</sub> contains ''T''<sub>1</sub>. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''weaker'''.
+;Finitely generated: See '''[[Alexandrov topology]]'''.
+;[[First category]]: See '''[[Meagre set|Meagre]]'''.
+;[[First-countable space|First-countable]]: A space is [[First-countable space|first-countable]] if every point has a [[countable]] local base.
+;Fréchet: See '''T<sub>1</sub>'''.
+;Frontier: See '''[[Boundary (topology)|Boundary]]'''.
+;Full set: A [[compact space|compact]] subset ''K'' of the [[complex plane]] is called '''full''' if its [[complement (set theory)|complement]] is connected. For example, the [[closed unit disk]] is full, while the [[unit circle]] is not.
+;Functionally separated: Two sets ''A'' and ''B'' in a space ''X'' are functionally separated if there is a continuous map ''f'': ''X'' &nbsp;→&nbsp; [0, 1] such that ''f''(''A'') = 0 and ''f''(''B'') = 1.
+== G ==
+;[[G-delta set|''G''<sub>δ</sub> set]]: A [[G-delta set|''G''<sub>δ</sub> set]] or '''inner limiting set''' is a [[countable]] intersection of open sets.<ref name=ss162>Steen &amp; Seebach (1978) p.162</ref>
+;''G''<sub>δ</sub> space: A space in which every closed set is a ''G''<sub>δ</sub> set.<ref name=ss162/>
+;[[Generic point]]: A [[generic point]] for a closed set is a point for which the closed set is the closure of the singleton set containing that point.<ref>Vickers (1989) p.65</ref>
+== H ==
+; [[Hausdorff space|Hausdorff]]: A [[Hausdorff space]] (or '''[[T2 space|T<sub>2</sub>]] space''') is one in which every two distinct points have [[Disjoint sets|disjoint]] neighbourhoods. Every Hausdorff space is T<sub>1</sub>.
+; [[H-closed space|H-closed]]: A space is H-closed, or '''Hausdorff closed''' or '''absolutely closed''', if it is closed in every Hausdorff space containing it.
+; Hereditarily ''P'': A space is hereditarily ''P'' for some property ''P'' if every subspace is also ''P''.
+; [[Hereditary property|Hereditary]]: A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.<ref name=ss4>Steen &amp; Seebach p.4</ref> For example, second-countability is a hereditary property.
+; [[Homeomorphism]]: If ''X'' and ''Y'' are spaces, a [[homeomorphism]] from ''X'' to ''Y'' is a [[bijection|bijective]] function ''f'' :&nbsp;''X''&nbsp;→&nbsp;''Y'' such that ''f'' and ''f''<sup>−1</sup> are continuous. The spaces ''X'' and ''Y'' are then said to be '''homeomorphic'''. From the standpoint of topology, homeomorphic spaces are identical.
+; [[Homogeneous space|Homogeneous]]: A space ''X'' is [[Homogeneous space|homogeneous]] if, for every ''x'' and ''y'' in ''X'', there is a homeomorphism ''f'' : ''X'' &nbsp;→&nbsp; ''X'' such that ''f''(''x'') = ''y''. Intuitively, the space looks the same at every point. Every [[topological group]] is homogeneous.
+; [[homotopic|Homotopic maps]]: Two continuous maps  ''f'', ''g'' : ''X'' &nbsp;→&nbsp; ''Y'' are [[homotopic]] (in ''Y'') if there is a continuous map ''H'' : ''X'' × [0, 1] &nbsp;→&nbsp; ''Y'' such that ''H''(''x'', 0) = ''f''(''x'') and ''H''(''x'', 1) = ''g''(''x'') for all ''x'' in ''X''. Here, ''X'' × [0, 1] is given the product topology. The function ''H'' is called a '''homotopy''' (in ''Y'') between ''f'' and ''g''.
+; Homotopy: See '''[[homotopic|Homotopic maps]]'''.
+; [[Hyperconnected space|Hyper-connected]]: A space is hyper-connected if no two non-empty open sets are disjoint<ref name=ss29/>  Every hyper-connected space is connected.<ref name=ss29/>
+== I ==
+; Identification map: See '''[[Quotient map]]'''.
+; [[Quotient space|Identification space]]: See '''[[Quotient space]]'''.
+; [[Indiscrete space]]: See '''[[Trivial topology]]'''.
+; [[Infinite-dimensional topology]]: See '''[[Hilbert manifold]]''' and '''[[Q-manifolds]]''', i.e. (generalized) manifolds modelled on the Hilbert space and on the Hilbert cube respectively.
+; Inner limiting set: A ''G''<sub>δ</sub> set.<ref name=ss162/>
+; [[Interior (topology)|Interior]]: The [[interior (topology)|interior]] of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set ''S'' is an '''interior point''' of ''S''.
+; Interior point: See '''[[Interior (topology)|Interior]]'''.
+; [[Isolated point]]: A point ''x'' is an [[isolated point]] if the [[singleton (mathematics)|singleton]] {''x''} is open. More generally, if ''S'' is a subset of a space ''X'', and if ''x'' is a point of ''S'', then ''x'' is an isolated point of ''S'' if {''x''} is open in the subspace topology on ''S''.
+; Isometric isomorphism: If ''M''<sub>1</sub> and ''M''<sub>2</sub> are metric spaces, an isometric isomorphism from ''M''<sub>1</sub> to ''M''<sub>2</sub> is a [[bijection|bijective]] isometry ''f'' : ''M''<sub>1</sub> &nbsp;→&nbsp; ''M''<sub>2</sub>. The metric spaces are then said to be '''isometrically isomorphic'''. From the standpoint of metric space theory, isometrically isomorphic spaces are identical.
+; Isometry: If (''M''<sub>1</sub>, ''d''<sub>1</sub>) and (''M''<sub>2</sub>, ''d''<sub>2</sub>) are metric spaces, an isometry from ''M''<sub>1</sub> to ''M''<sub>2</sub> is a function ''f'' : ''M''<sub>1</sub> &nbsp;→&nbsp; ''M''<sub>2</sub> such that ''d''<sub>2</sub>(''f''(''x''), ''f''(''y'')) = ''d''<sub>1</sub>(''x'', ''y'') for all ''x'', ''y'' in ''M''<sub>1</sub>. Every isometry is [[Injective function|injective]], although not every isometry is [[surjection|surjective]].
+== K ==
+;[[Kolmogorov space|Kolmogorov axiom]]: See '''[[T0 space|T<sub>0</sub>]]'''.
+;[[Kuratowski closure axioms]]: The [[Kuratowski closure axioms]] is a set of [[axiom]]s satisfied by the function which takes each subset of ''X'' to its closure:
+:# ''[[Isotone function|Isotonicity]]'': Every set is contained in its closure.
+:# ''[[Idempotent function|Idempotence]]'': The closure of the closure of a set is equal to the closure of that set.
+:# ''Preservation of binary unions'': The closure of the union of two sets is the union of their closures.
+:# ''Preservation of nullary unions'': The closure of the empty set is empty.
+:If ''c'' is a function from the [[power set]] of ''X'' to itself, then ''c'' is a '''closure operator''' if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on ''X'' by declaring the closed sets to be the [[fixed point (mathematics)|fixed point]]s of this operator, i.e. a set ''A'' is closed [[if and only if]] ''c''(''A'') = ''A''.
+== L ==
+;L-space: An ''L-space'' is a [[Hereditary property#In topology|hereditarily]] [[Lindelöf space|Lindelöf]] space which is not hereditarily [[Separable space|separable]].  A [[Suslin line]] would be an L-space.<ref name=GKW290>{{cite book | title=Sets and Extensions in the Twentieth Century | editor1-first=Dov M. | editor1-last=Gabbay | editor2-first=Akihiro | editor2-last=Kanamori | editor3-first=John Hayden | editor3-last=Woods | publisher=Elsevier | year=2012 | isbn=0444516212 | page=290 }}</ref>
+;Larger topology: See '''[[Finer topology]]'''.
+;[[Limit point]]: A point ''x'' in a space ''X'' is a [[limit point]] of a subset ''S'' if every open set containing ''x'' also contains a point of ''S'' other than ''x'' itself. This is equivalent to requiring that every neighbourhood of ''x'' contains a point of ''S'' other than ''x'' itself.
+;Limit point compact: See '''Weakly countably compact'''.
+;[[Lindelöf space|Lindelöf]]: A space is [[Lindelöf space|Lindelöf]] if every open cover has a [[countable]] subcover.
+;[[Local base]]: A set ''B'' of neighbourhoods of a point ''x'' of a space ''X'' is a local base (or '''local basis''', '''neighbourhood base''', '''neighbourhood basis''') at ''x'' if every neighbourhood of ''x'' contains some member of ''B''.
+;Local basis: See '''Local base'''.
+;Locally closed subset: A subset of a topological space that is the intersection of an open and a closed subset.  Equivalently, it is a relatively open subset of its closure.
+;[[Locally compact space|Locally compact]]: A space is [[Locally compact space|locally compact]] if every point has a local base consisting of compact neighbourhoods. Every locally compact Hausdorff space is Tychonoff.
+;[[Locally connected]]: A space is [[locally connected]] if every point has a local base consisting of connected neighbourhoods.
+; [[Locally finite collection|Locally finite]]: A collection of subsets of a space is [[Locally finite collection|locally finite]] if every point has a neighbourhood which has nonempty intersection with only [[finite set|finite]]ly many of the subsets.  See also '''countably locally finite''', '''[[point finite]]'''.
+;Locally metrizable'''/'''Locally metrisable: A space is locally metrizable if every point has a metrizable neighbourhood.
+;[[Locally path-connected]]: A space is [[locally path-connected]] if every point has a local base consisting of path-connected neighbourhoods. A locally path-connected space is connected [[if and only if]] it is path-connected.
+;Locally simply connected: A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods.
+;[[Loop (topology)|Loop]]: If ''x'' is a point in a space ''X'', a [[loop (topology)|loop]] at ''x'' in ''X'' (or a loop in ''X'' with basepoint ''x'') is a path ''f'' in ''X'', such that ''f''(0) = ''f''(1) = ''x''. Equivalently, a loop in ''X'' is a continuous map from the [[unit circle]] ''S''<sup>1</sup> into ''X''.
+== M ==
+;[[Meagre set|Meagre]]: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is meagre in ''X'' (or of '''first category''' in ''X'') if it is the [[countable]] union of nowhere dense sets.  If ''A'' is not meagre in ''X'', ''A'' is of '''second category''' in ''X''.<ref name=ss7>Steen &amp; Seebach (1978) p.7</ref>
+;Metric: See '''[[Metric space]]'''.
+;Metric invariant: A metric invariant is a property which is preserved under isometric isomorphism.
+;[[Metric map]]: If ''X'' and ''Y'' are metric spaces with metrics ''d''<sub>''X''</sub> and ''d''<sub>''Y''</sub> respectively, then a [[metric map]] is a function ''f'' from ''X'' to ''Y'', such that for any points ''x'' and ''y'' in ''X'', ''d''<sub>''Y''</sub>(''f''(''x''), ''f''(''y'')) ≤ ''d''<sub>''X''</sub>(''x'', ''y''). A metric map is [[metric map|strictly metric]] if the above inequality is strict for all ''x'' and ''y'' in ''X''.
+;[[Metric space]]: A [[metric space]] (''M'', ''d'') is a set ''M'' equipped with a function ''d'' :&nbsp;''M''&nbsp;×&nbsp;''M''&nbsp;→&nbsp;'''[[real number|R]]''' satisfying the following axioms for all ''x'', ''y'', and ''z'' in ''M'':
+:# ''d''(''x'', ''y'') ≥ 0
+:# ''d''(''x'', ''x'') = 0
+:# if &nbsp; ''d''(''x'', ''y'') = 0 &nbsp; then &nbsp; ''x'' = ''y'' &nbsp;&nbsp;&nbsp; (''identity of indiscernibles'')
+:# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')  &nbsp;&nbsp;&nbsp; (''symmetry'')
+:# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') &nbsp;&nbsp;&nbsp;  (''[[triangle inequality]]'')
+:The function ''d'' is a '''metric''' on ''M'', and ''d''(''x'', ''y'') is the '''distance''' between ''x'' and ''y''. The collection of all open balls of ''M'' is a base for a topology on ''M''; this is the topology on ''M'' induced by ''d''. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable.
+;[[Metrizable]]'''/'''Metrisable: A space is [[metrizable]] if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable.
+;Monolith: Every non-empty ultra-connected compact space ''X'' has a largest proper open subset; this subset is called a '''monolith'''.
+;[[Moore space (topology)|Moore space]]: A [[Moore space (topology)|Moore space]] is a [[developable space|developable]] [[regular Hausdorff space]].<ref name=ss163>Steen &amp; Seebach (1978) p.163</ref>
+== N ==
+;[[Neighbourhood (mathematics)|Neighbourhood]]'''/'''Neighborhood: A neighbourhood of a point ''x'' is a set containing an open set which in turn contains the point ''x''. More generally, a neighbourhood of a set ''S'' is a set containing an open set which in turn contains the set ''S''. A neighbourhood of a point ''x'' is thus a neighbourhood of the [[singleton (mathematics)|singleton]] set {''x''}. (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.)
+;[[Local base|Neighbourhood base]]/basis: See '''[[Local base]]'''.
+;Neighbourhood system for a point ''x'': A [[neighbourhood system]] at a point ''x'' in a space is the collection of all neighbourhoods of ''x''.
+;[[Net (mathematics)|Net]]: A [[net (mathematics)|net]] in a space ''X'' is a map from a [[directed set]] ''A'' to ''X''. A net from ''A'' to ''X'' is usually denoted (''x''<sub>α</sub>), where α is an [[index set|index variable]] ranging over ''A''. Every [[sequence]] is a net, taking ''A'' to be the directed set of [[natural number]]s with the usual ordering.
+;[[Normal space|Normal]]: A space is [[normal space|normal]] if any two disjoint closed sets have disjoint neighbourhoods.<ref name=ss162/>  Every normal space admits a partition of unity.
+;[[T4 space|Normal Hausdorff]]: A [[T4 space|normal Hausdorff]] space (or [[T4 space|'''T<sub>4</sub>''' space]]) is a normal T<sub>1</sub> space. (A normal space is Hausdorff [[if and only if]] it is T<sub>1</sub>, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.
+;[[Nowhere dense set|Nowhere dense]]: A [[nowhere dense set]] is a set whose closure has empty interior.
+== O ==
+; [[Open cover]]: An [[open cover]] is a cover consisting of open sets.<ref name=ss163/>
+; Open ball: If (''M'', ''d'') is a metric space, an open ball is a set of the form ''B''(''x''; ''r'') := {''y'' in ''M'' : ''d''(''x'', ''y'') < ''r''}, where ''x'' is in ''M'' and ''r'' is a [[positive number|positive]] [[real number]], the '''radius''' of the ball. An open ball of radius ''r'' is an '''open ''r''-ball'''. Every open ball is an open set in the topology on ''M'' induced by ''d''.
+; Open condition:  See '''open property'''.
+; [[Open set]]: An [[open set]] is a member of the topology.
+; [[Open map|Open function]]: A function from one space to another is [[open map|open]] if the [[image (mathematics)|image]] of every open set is open.
+; Open property:  A property of points in a [[topological space]] is said to be "open" if those points which possess it form an [[open set]].  Such conditions often take a common form, and that form can be said to be an ''open condition''; for example, in [[metric space]]s, one defines an open ball as above, and says that "strict inequality is an open condition".
+== P ==<!-- This section is linked from [[Closure (topology)]] -->
+;[[Paracompact space|Paracompact]]: A space is [[paracompact space|paracompact]] if every open cover has a locally finite open refinement.  Paracompact implies metacompact.<ref name=ss23>Steen &amp; Seebach (1978) p.23</ref>  Paracompact Hausdorff spaces are normal.<ref name=ss25>Steen &amp; Seebach (1978) p.25</ref>
+;[[Partition of unity]]: A partition of unity of a space ''X'' is a set of continuous functions from ''X'' to [0, 1] such that any point has a neighbourhood where all but a [[finite set|finite]] number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.
+;[[Path (topology)|Path]]: A [[Path (topology)|path]] in a space ''X'' is a continuous map ''f'' from the closed unit [[interval (mathematics)|interval]] [0, 1] into ''X''. The point ''f''(0) is the initial point of ''f''; the point ''f''(1) is the terminal point of ''f''.<ref name=ss29>Steen &amp; Seebach (1978) p.29</ref>
+;[[Path-connected space|Path-connected]]: A space ''X'' is [[path-connected space|path-connected]] if, for every two points ''x'', ''y'' in ''X'', there is a path ''f'' from ''x'' to ''y'', i.e., a path with initial point ''f''(0) = ''x'' and terminal point ''f''(1) = ''y''. Every path-connected space is connected.<ref name=ss29/>
+;Path-connected component: A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a [[partition of a set|partition]] of that space, which is [[partition of a set|finer]] than the partition into connected components.<ref name=ss29/> The set of path-connected components of a space ''X'' is denoted [[homotopy groups|π<sub>0</sub>(''X'')]].
+;Perfectly normal: a normal space which is also a G<sub>δ</sub>.<ref name=ss162/>
+;π-base: A collection ''B'' of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includes a set from ''B''.<ref>Hart, Nagata, Vaughan Sect. d-22, page 227</ref>
+;Point: A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point".
+;Point of closure: See '''[[Closure (topology)|Closure]]'''.
+;[[Polish space|Polish]]: A space is Polish if it is separable and topologically complete, i.e. if it is homeomorphic to a separable and complete metric space.
+;P-point: A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersections.
+;Pre-compact: See '''[[Relatively compact]]'''.
+;[[Product topology]]: If {''X''<sub>''i''</sub>} is a collection of spaces and ''X'' is the (set-theoretic) [[Cartesian product|product]] of {''X''<sub>''i''</sub>}, then the [[product topology]] on ''X'' is the coarsest topology for which all the projection maps are continuous.
+;Proper function/mapping: A continuous function ''f'' from a space ''X'' to a space ''Y'' is proper if ''f''<sup>−1</sup>(''C'') is a compact set in ''X'' for any compact subspace ''C'' of ''Y''.
+;[[Proximity space]]: A proximity space (''X'',&nbsp;'''δ''') is a set ''X'' equipped with a [[binary relation]] '''δ''' between subsets of ''X'' satisfying the following properties:
+:For all subsets ''A'', ''B'' and ''C'' of ''X'',
+:#''A'' '''δ''' ''B'' implies ''B'' '''δ''' ''A''
+:#''A'' '''δ''' ''B'' implies ''A'' is non-empty
+:#If ''A'' and ''B'' have non-empty intersection, then ''A'' '''δ''' ''B''
+:#''A'' '''δ''' (''B''&nbsp;∪&nbsp;''C'') [[iff]] (''A'' '''δ''' ''B'' or ''A'' '''δ''' ''C'')
+:#If, for all subsets ''E'' of ''X'', we have (''A'' '''δ''' ''E'' or ''B'' '''δ''' ''E''), then we must have ''A'' '''δ''' (''X'' − ''B'')
+;Pseudocompact: A space is pseudocompact if every [[real number|real-valued]] continuous function on the space is bounded.
+;Pseudometric: See '''Pseudometric space'''.
+;Pseudometric space: A pseudometric space (''M'', ''d'') is a set ''M'' equipped with a function ''d'' :&nbsp;''M''&nbsp;×&nbsp;''M''&nbsp;→&nbsp;'''[[real number|R]]''' satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical.  The function ''d'' is a '''pseudometric''' on ''M''. Every metric is a pseudometric.
+;Punctured neighbourhood'''/'''Punctured neighborhood: A punctured neighbourhood of a point ''x'' is a neighbourhood of ''x'', minus {''x''}. For instance, the [[interval (mathematics)|interval]] (−1, 1) = {''y'' : −1 < ''y'' < 1} is a neighbourhood of ''x'' = 0 in the [[real line]], so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0.
+== Q ==
+;Quasicompact: See '''[[compact space|compact]]'''. Some authors define "compact" to include the [[Hausdorff space|Hausdorff]] separation axiom, and they use the term '''quasicompact''' to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French.
+;Quotient map: If ''X'' and ''Y'' are spaces, and if ''f'' is a [[surjection]] from ''X'' to ''Y'', then ''f'' is a quotient map (or '''identification map''') if, for every subset ''U'' of ''Y'', ''U'' is open in ''Y'' [[if and only if]] ''f''<sup>&nbsp;<tt>-</tt>1</sup>(''U'') is open in ''X''.  In other words, ''Y'' has the ''f''-strong topology.  Equivalently, <math>f</math> is a quotient map if and only if it is the transfinite composition of maps <math>X\rightarrow X/Z</math>, where <math>Z\subset X</math> is a subset. Note that this doesn't imply that ''f'' is an open function.
+;[[Quotient space]]: If ''X'' is a space, ''Y'' is a set, and ''f'' :&nbsp;''X''&nbsp;→&nbsp;''Y'' is any [[surjection|surjective]] function, then the [[quotient space|quotient topology]] on ''Y'' induced by ''f'' is the finest topology for which ''f'' is continuous. The space ''X'' is a quotient space or '''identification space'''. By definition, ''f'' is a quotient map. The most common example of this is to consider an [[equivalence relation]] on ''X'', with ''Y'' the set of [[equivalence class]]es and ''f'' the natural projection map. This construction is dual to the construction of the subspace topology.
+== R ==
+; Refinement: A cover ''K'' is a [[refinement (topology)|refinement]] of a cover ''L'' if every member of ''K'' is a subset of some member of ''L''.
+; [[Regular space|Regular]]: A space is [[regular space|regular]] if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and ''x'' have [[Disjoint sets|disjoint]] neighbourhoods.
+; [[T3 space|Regular Hausdorff]]: A space is [[T3 space|regular Hausdorff]] (or '''T<sub>3</sub>''') if it is a regular T<sub>0</sub> space.  (A regular space is Hausdorff [[if and only if]] it is T<sub>0</sub>, so the terminology is consistent.)
+; Regular open: An open subset ''U'' of a space ''X'' is regular open if it equals the interior of its closure; simiarly, a regular closed set is equal to the closure of is interior.<ref name="Steen & Seebach 1978 p.6">Steen & Seebach (1978) p.6</ref>  An example of a non-regular open set is the set  ''U'' = {{open-open|0,1}} ∪ {{open-open|1,2}} in '''R''' with its normal topology, since 1 is in the interior of the closure of ''U'', but not in ''U''. The regular open subsets of a space form a [[complete Boolean algebra]].<ref name="Steen & Seebach 1978 p.6"/>
+; [[Relatively compact]]: A subset ''Y'' of a space ''X'' is [[relatively compact]] in ''X'' if the closure of ''Y'' in ''X'' is compact.
+; Residual: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is residual in ''X'' if the complement of ''A'' is meagre in ''X''. Also called '''comeagre''' or '''comeager'''.
+; Resolvable: A [[topological space]] is called [[resolvable space|resolvable]] if it is expressible as the union of two [[disjoint sets|disjoint]] [[dense subset]]s.
+; Rim-compact: A space is rim-compact if it has a base of open sets whose boundaries are compact.
+== S ==
+;[[S-space]]:  An ''S-space'' is a [[Hereditary property#In topology|hereditarily]] [[Separable space|separable]] space which is not hereditarily [[Lindelöf space|Lindelöf]].<ref name=GKW290/>
+;Scattered: A space ''X'' is scattered if every nonempty subset ''A'' of ''X'' contains a point isolated in ''A''.
+;[[Scott continuity|Scott]]: The [[Scott topology]] on a [[poset]] is that in which the open sets are those [[Upper set]]s inaccessible by directed joins.<ref>Vickers (1989) p.95</ref>
+;Second category: See '''Meagre'''.
+;[[Second-countable space|Second-countable]]: A space is [[second-countable space|second-countable]] or '''perfectly separable''' if it has a [[countable]] base for its topology.<ref name=ss162/> Every second-countable space is first-countable, separable, and Lindelöf.
+;[[Semilocally simply connected]]: A space ''X'' is [[semilocally simply connected]] if, for every point ''x'' in ''X'', there is a neighbourhood ''U'' of ''x'' such that every loop at ''x'' in ''U'' is homotopic in ''X'' to the constant loop ''x''. Every simply connected space and every locally simply connected space is semilocally simply connected. (Compare with locally simply connected; here, the homotopy is allowed to live in ''X'', whereas in the definition of locally simply connected, the homotopy must live in ''U''.)
+;[[semiregular space|Semiregular]]: A space is semiregular if the regular open sets form a base.
+;[[Separable (topology)|Separable]]: A space is [[separable (topology)|separable]] if it has a [[countable]] dense subset.<ref name=ss162/><ref name=ss7/>
+;[[Separated sets|Separated]]: Two sets ''A'' and ''B'' are [[separated sets|separated]] if each is [[Disjoint sets|disjoint]] from the other's closure.
+;Sequentially compact: A space is sequentially compact if every [[sequence]] has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.
+;[[Short map]]: See '''[[metric map]]'''
+;[[Simply connected space|Simply connected]]: A space is [[simply connected space|simply connected]] if it is path-connected and every loop is homotopic to a constant map.
+;Smaller topology: See '''[[Coarser topology]]'''.
+;[[Sober space|Sober]]: In a [[sober space]], every [[hyperconnected space|irreducible]] closed subset is the [[closure (topology)|closure]] of exactly one point: that is, has a unique [[generic point]].<ref>Vickers (1989) p.66</ref>
+;Star: The star of a point in a given [[cover (topology)|cover]] of a [[topological space]] is the union of all the sets in the cover that contain the point. See '''[[star refinement]]'''.
+;<math>f</math>-Strong topology:  Let <math>f\colon X\rightarrow Y</math> be a map of topological spaces.  We say that <math>Y</math> has the <math>f</math>-strong topology if, for every subset <math>U\subset Y</math>, one has that <math>U</math> is open in <math>Y</math> if and only if <math>f^{-1}(U)</math> is open in <math>X</math>
+;Stronger topology: See '''[[Finer topology]]'''. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''weaker topology'''.
+;[[Subbase]]: A collection of open sets is a [[subbase]] (or '''subbasis''') for a topology if every non-empty proper open set in the topology is a union of [[finite set|finite]] intersections of sets in the subbase. If ''B'' is ''any'' collection of subsets of a set ''X'', the topology on ''X'' generated by ''B'' is the smallest topology containing ''B''; this topology consists of the empty set, ''X'' and all unions of finite intersections of elements of ''B''.
+;[[Subbase|Subbasis]]: See '''[[Subbase]]'''.
+;Subcover: A cover ''K'' is a subcover (or '''subcovering''') of a cover ''L'' if every member of ''K'' is a member of ''L''.
+;Subcovering: See '''Subcover'''.
+;Submaximal space: A [[topological space]] is said to be '''submaximal''' if every subset of it is locally closed, that is, every subset is the intersection of an [[open set]] and a [[closed set]].
+Here are some facts about submaximality as a property of topological spaces:
+* Every [[door space]] is submaximal.
+* Every submaximal space is ''weakly submaximal'' viz every finite set is locally closed.
+* Every submaximal space is [[irresolvable space|irresolvable]]<ref>{{citation | title=Recent progress in general topology | volume=2 | series=Recent Progress in General Topology | author1=Miroslav Hušek | author2=J. van Mill | publisher=Elsevier | year=2002 | isbn=0-444-50980-1 | page=21 }}</ref>
+;Subspace: If ''T'' is a topology on a space ''X'', and if ''A'' is a subset of ''X'', then the [[subspace topology]] on ''A'' induced by ''T'' consists of all intersections of open sets in ''T'' with ''A''. This construction is dual to the construction of the quotient topology.
+== T ==
+;[[T0 space|T<sub>0</sub>]]: A space is [[T0 space|T<sub>0</sub>]] (or '''Kolmogorov''') if for every pair of distinct points ''x'' and ''y'' in the space, either there is an open set containing ''x'' but not ''y'', or there is an open set containing ''y'' but not ''x''.
+;[[T1 space|T<sub>1</sub>]]: A space is [[T1 space|T<sub>1</sub>]] (or '''Fréchet''' or '''accessible''') if for every pair of distinct points ''x'' and ''y'' in the space, there is an open set containing ''x'' but not ''y''. (Compare with T<sub>0</sub>; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T<sub>1</sub> if all its [[singleton (mathematics)|singleton]]s are closed. Every T<sub>1</sub> space is T<sub>0</sub>.
+;[[T2 space|T<sub>2</sub>]]: See '''[[Hausdorff space]]'''.
+;[[T3 space|T<sub>3</sub>]]: See '''[[T3 space|Regular Hausdorff]]'''.
+;[[Tychonoff space|T<sub>3½</sub>]]: See '''[[Tychonoff space]]'''.
+;[[T4 space|T<sub>4</sub>]]: See '''[[T4 space|Normal Hausdorff]]'''.
+;[[T5 space|T<sub>5</sub>]]: See '''[[T5 space|Completely normal Hausdorff]]'''.
+;[[Category of topological spaces|Top]]: See '''[[Category of topological spaces]]'''.
+;[[Topological invariant]]: A topological invariant is a property which is preserved under homeomorphism. For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. [[Algebraic topology]] is the study of topologically invariant [[abstract algebra]] constructions on topological spaces.
+;[[Topological space]]: A [[topological space]] (''X'', ''T'') is a set ''X'' equipped with a collection ''T'' of subsets of ''X'' satisfying the following [[axiom]]s:
+:# The empty set and ''X'' are in ''T''.
+:# The union of any collection of sets in ''T'' is also in ''T''.
+:# The intersection of any pair of sets in ''T'' is also in ''T''.
+:The collection ''T'' is a '''topology''' on ''X''.
+;Topological sum: See '''Coproduct topology'''.
+;[[Complete space|Topologically complete]]: A space is [[complete space|topologically complete]] if it is homeomorphic to a complete metric space.
+;Topology: See '''[[Topological space]]'''.
+;Totally bounded: A metric space ''M'' is totally bounded if, for every ''r'' > 0, there exist a [[finite set|finite]] cover of ''M'' by open balls of radius ''r''. A metric space is compact if and only if it is complete and totally bounded.
+;Totally disconnected: A space is totally disconnected if it has no connected subset with more than one point.
+;[[Trivial topology]]: The [[trivial topology]] (or '''indiscrete topology''') on a set ''X'' consists of precisely the empty set and the entire space ''X''.
+;[[Tychonoff space|Tychonoff]]: A [[Tychonoff space]] (or '''completely regular Hausdorff''' space, '''completely T<sub>3</sub>''' space, '''T<sub>3.5</sub>''' space) is a completely regular T<sub>0</sub> space.  (A completely regular space is Hausdorff [[if and only if]] it is T<sub>0</sub>, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.
+== U ==
+;Ultra-connected: A space is ultra-connected if no two non-empty closed sets are disjoint.<ref name="ss29"/> Every ultra-connected space is path-connected.
+;[[Ultrametric space|Ultrametric]]: A metric is an ultrametric if it satisfies the following stronger version of the [[triangle inequality]]: for all ''x'', ''y'', ''z'' in ''M'', ''d''(''x'', ''z'') ≤ max(''d''(''x'', ''y''), ''d''(''y'', ''z'')).
+;[[Uniform isomorphism]]: If ''X'' and ''Y'' are [[uniform space]]s, a uniform isomorphism from ''X'' to ''Y'' is a bijective function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''<sup>−1</sup> are [[uniformly continuous]]. The spaces are then said to be uniformly isomorphic and share the same [[uniform properties]].
+;[[Uniformizable]]/Uniformisable: A space is uniformizable if it is homeomorphic to a uniform space.
+;[[Uniform space]]: A [[uniform space]] is a set ''U'' equipped with a nonempty collection Φ of subsets of the [[Cartesian product]] ''X'' × ''X'' satisfying the following [[axiom]]s:
+:# if ''U'' is in Φ, then ''U'' contains { (''x'', ''x'') | ''x'' in ''X'' }.
+:# if ''U'' is in Φ, then { (''y'', ''x'') | (''x'', ''y'') in ''U'' } is also in Φ
+:# if ''U'' is in Φ and ''V'' is a subset of ''X'' × ''X'' which contains ''U'', then ''V'' is in Φ
+:# if ''U'' and ''V'' are in Φ, then ''U'' ∩ ''V'' is in Φ
+:# if ''U'' is in Φ, then there exists ''V'' in Φ such that, whenever (''x'', ''y'') and (''y'', ''z'') are in ''V'', then (''x'', ''z'') is in ''U''.
+:The elements of Φ are called '''entourages''', and Φ itself is called a '''uniform structure''' on ''U''.
+;Uniform structure: See '''[[Uniform space]]'''.
+== W ==
+; [[Weak topology]]: The [[weak topology]] on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
+; Weaker topology: See '''[[Coarser topology]]'''. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''stronger topology'''.
+; Weakly countably compact: A space is weakly countably compact (or '''limit point compact''') if every [[Infinity|infinite]] subset has a limit point.
+; Weakly hereditary: A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.
+; Weight: The [[weight of a space]] ''X'' is the smallest [[cardinal number]] κ such that ''X'' has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is [[well order|well-ordered]].)
+; Well-connected: See '''Ultra-connected'''. (Some authors use this term strictly for ultra-connected compact spaces.)
+== Z ==
+;Zero-dimensional: A space is [[zero-dimensional]] if it has a base of clopen sets.<ref name=ss33>Steen &amp; Seebach (1978) p.33</ref>
+== References ==
+{{reflist}}
+* {{cite book | title=Encyclopedia of general topology | first1=Klaas Pieter | last1=Hart | first2=Jun-iti | last2=Nagata | first3=Jerry E. | last3=Vaughan | publisher=Elsevier | year=2004 | isbn=978-0-444-50355-8 }}
+*{{cite book|title=Handbook of Set-Theoretic Topology|last=Kunen|first=Kenneth|authorlink=Kenneth Kunen|coauthors=Vaughan, Jerry E. (''editors'')|publisher=North-Holland|isbn=0-444-86580-2}}
+*{{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 }}
+*{{cite book | first=Steven | last=Vickers | authorlink=Steve Vickers (computer scientist) | title=Topology via Logic | series=Cambridge Tracts in Theoretic Computer Science | volume=5 | isbn=0-521-36062-5 | year=1989 | zbl=0668.54001 }}
+*{{cite book |first=Stephen |last=Willard |title=General Topology |publisher=Addison-Wesley| year=1970 |isbn=978-0-201-08707-9}} Also available as Dover reprint.
+== External links ==
+* [http://math.berkeley.edu/~apollo/topodefs.ps A glossary of definitions in topology]
+{{DEFAULTSORT:Glossary Of Topology}}
+[[Category:Properties of topological spaces]]
+[[Category:Glossaries of mathematics|Topology]]
+[[Category:General topology| ]]
+[[Category:Algebraic topology| ]]
+[[Category:Geometric topology| ]]
+[[Category:Differential topology| ]]

Triple point: Difference between revisions

Revision as of 13:18, 30 October 2013

A

B

C

D

E

F

G

H

I

K

L

M

N

O

P

Q

R

S

T

U

W

Z

References

External links

Navigation menu

Search