Difference between revisions of "Final topology"
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Revision as of 14:01, 4 March 2014
In general topology and related areas of mathematics, the final topology (or strong topology or colimit topology or projective topology) on a set , with respect to a family of functions into , is the finest topology on X which makes those functions continuous.
The dual notion is the initial topology.
Definition
Given a set and a family of topological spaces with functions
the final topology on is the finest topology such that each
is continuous.
Explicitly, the final topology may be described as follows: a subset U of X is open if and only if is open in Y_{i} for each i ∈ I.
Examples
 The quotient topology is the final topology on the quotient space with respect to the quotient map.
 The disjoint union is the final topology with respect to the family of canonical injections.
 More generally, a topological space is coherent with a family of subspaces if it has the final topology coinduced by the inclusion maps.
 The direct limit of any direct system of spaces and continuous maps is the settheoretic direct limit together with the final topology determined by the canonical morphisms.
 Given a family of topologies {τ_{i}} on a fixed set X the final topology on X with respect to the functions id_{X} : (X, τ_{i}) → X is the infimum (or meet) of the topologies {τ_{i}} in the lattice of topologies on X. That is, the final topology τ is the intersection of the topologies {τ_{i}}.
 The etale space of a sheaf is topologized by a final topology.
Properties
A subset of is closed/open if and only if its preimage under f_{i} is closed/open in for each i ∈ I.
The final topology on X can be characterized by the following universal property: a function from to some space is continuous if and only if is continuous for each i ∈ I.
By the universal property of the disjoint union topology we know that given any family of continuous maps f_{i} : Y_{i} → X there is a unique continuous map
If the family of maps f_{i} covers X (i.e. each x in X lies in the image of some f_{i}) then the map f will be a quotient map if and only if X has the final topology determined by the maps f_{i}.
Categorical description
In the language of category theory, the final topology construction can be described as follows. Let Y be a functor from a discrete category J to the category of topological spaces Top which selects the spaces Y_{i} for i in J. Let Δ be the diagonal functor from Top to the functor category Top^{J} (this functor sends each space X to the constant functor to X). The comma category (Y ↓ Δ) is then the category of cones from Y, i.e. objects in (Y ↓ Δ) are pairs (X, f) where f_{i} : Y_{i} → X is a family of continuous maps to X. If U is the forgetful functor from Top to Set and Δ′ is the diagonal functor from Set to Set^{J} then the comma category (UY ↓ Δ′) is the category of all cones from UY. The final topology construction can then be described as a functor from (UY ↓ Δ′) to (Y ↓ Δ). This functor is left adjoint to the corresponding forgetful functor.
See also
References
 Stephen Willard, General Topology, (1970) AddisonWesley Publishing Company, Reading Massachusetts. (Provides a short, general introduction)