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In [[mathematics]], a '''pseudometric space''' is a generalized [[metric space]] in which the distance between two distinct points can be zero. In the same way as every [[normed space]] is a [[metric space]], every [[seminormed space]] is a pseudometric space. Because of this analogy the term [[semimetric space]] (which has a different meaning in [[topology]]) is sometimes used as a synonym, especially in [[functional analysis]].


When a topology is generated using a family of pseudometrics, the space is called a [[gauge space]].


==Definition==
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A pseudometric space <math>(X,d)</math> is a set <math>X</math> together with a non-negative real-valued function <math>d: X \times X \longrightarrow \mathbb{R}_{\geq 0}</math> (called a '''pseudometric''') such that, for every <math>x,y,z \in X</math>,
 
#<math>\,\!d(x,x) = 0</math>.
#<math>\,\!d(x,y) = d(y,x)</math> (''symmetry'')
#<math>\,\!d(x,z) \leq d(x,y) + d(y,z)</math> (''[[subadditivity]]''/''[[triangle inequality]]'')
<!-- Leave those  \,\!  at the start of each condition; see WP:MATH for why. -->
 
Unlike a metric space, points in a pseudometric space need not be [[identity of indiscernibles|distinguishable]]; that is, one may have <math>d(x,y)=0</math> for distinct values <math>x\ne y</math>.
 
==Examples==
Pseudometrics arise naturally in [[functional analysis]]. Consider the space <math>\mathcal{F}(X)</math> of real-valued functions <math>f:X\to\mathbb{R}</math> together with a special point <math>x_0\in X</math>. This point then induces a pseudometric on the space of functions, given by
 
:<math>\,\!d(f,g) = |f(x_0)-g(x_0)|\;</math>
 
for <math>f,g\in \mathcal{F}(X)</math>
 
For vector spaces ''V'', a [[seminorm]] ''p'' induces a pseudometric on ''V'', as
 
:<math>\,\!d(x,y)=p(x-y).</math>
 
Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.
 
Pseudometrics also arise in the theory of hyperbolic [[complex manifold]]s: see [[Kobayashi metric]].
 
==Topology==
The '''pseudometric topology''' is the [[topological space|topology]] induced by the [[open balls]]
 
:<math>B_r(p)=\{ x\in X\mid d(p,x)<r \},</math>
 
which form a [[basis (topology)|basis]] for the topology.<ref>{{planetmath reference|id=6284|title=Pseudometric topology}}</ref> A topological space is said to be a '''pseudometrizable topological space''' if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.
 
The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is [[T0 space|T<sub>0</sub>]] (i.e. distinct points are topologically distinguishable).
 
==Metric identification==
The vanishing of the pseudometric induces an [[equivalence relation]], called the '''metric identification''', that converts the pseudometric space into a full-fledged [[metric space]]. This is done by defining <math>x\sim y</math> if <math>d(x,y)=0</math>. Let <math>X^*=X/{\sim}</math> and let
:<math>d^*([x],[y])=d(x,y)</math>
Then <math>d^*</math> is a metric on <math>X^*</math> and <math>(X^*,d^*)</math> is a well-defined metric space.<ref>{{cite book|last=Howes|first=Norman R.|title=Modern Analysis and Topology|year=1995|publisher=Springer|location=New York, NY|isbn=0-387-97986-7|url=http://www.springer.com/mathematics/analysis/book/978-0-387-97986-1|accessdate=10 September 2012|page=27|quote=Let <math>(X,d)</math> be a pseudo-metric space and define an equivalence relation <math>\sim</math> in <math>X</math> by <math>x \sim y</math> if <math>d(x,y)=0</math>. Let <math>Y</math> be the quotient space <math>X/\sim</math> and <math>p:X\to Y</math> the canonical projection that maps each point of <math>X</math> onto the equivalence class that contains it. Define the metric <math>\rho</math> in <math>Y</math> by <math>\rho(a,b) = d(p^{-1}(a),p^{-1}(b))</math> for each pair <math>a,b \in Y</math>. It is easily shown that <math>\rho</math> is indeed a metric and <math>\rho</math> defines the quotient topology on <math>Y</math>.}}</ref>
 
The metric identification preserves the induced topologies. That is, a subset <math>A\subset X</math> is open (or closed) in <math>(X,d)</math> if and only if <math>\pi(A)=[A]</math> is open (or closed) in <math>(X^*,d^*)</math>.
 
An example of this construction is the [[Complete_metric_space#Completion|completion of a metric space]] by its [[Cauchy sequences]].
 
==Notes==
{{Reflist}}
 
==References==
* {{cite book | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last=Arkhangel'skii | first=A.V. | coauthors=Pontryagin, L.S. | year=1990 | isbn=3-540-18178-4 | publisher=[[Springer Science+Business Media|Springer]] | series=Encyclopaedia of Mathematical Sciences}}
* {{cite book | title=Counterexamples in Topology | last=Steen | first=Lynn Arthur | coauthors=Seebach, Arthur | year=1995 | origyear=1970 | isbn=0-486-68735-X | publisher=[[Dover Publications]] | edition=new edition }}
* {{PlanetMath attribution|id=6273|title=Pseudometric space}}
* {{planetmath reference|id=6275|title=Example of pseudometric space}}
 
{{DEFAULTSORT:Pseudometric Space}}
[[Category:Properties of topological spaces]]
[[Category:Metric geometry]]

Revision as of 00:32, 27 February 2014


Even middle class and small class people cannot remain untouched with this problem. Some in the equipment helps with monitoring places and people remotely; via internet. CCTV security products that offers good potential for service providers in this segment. The topic is again, keep in mind, drenched in controversy and contorted through the gender biases of male verses female sexual stereotypes.

The 7 inch LCD screen to notice the cameras is available using the SW246 and SW3-DPS 4-Channel package. Ignore your audience: Although the Super Bowl does attract near to 100 million viewers, the crowd is more targeted that one would think. VCR's required regular maintenance and replacing VHS tape. Unitech Crestview has excellent connectivity to NH 8, so because of this to Delhi's official and commercial centres. This app provides quick access to your Android settings and enables you to switch to important menu items and control battery consumption. Here we're going to describe one hidden trend that is affecting the market industry as we speak.

Not only would you need to generate sure you might have suitable anti-virus software installed but you need to make sure it's kept approximately date. The 2600 4 and 4 channel DVR is expandable around 2TB the 16 channel is expandable as much as 4TB.

Each has its own set of benefits and it can be up for you which one you find for your car. They are very small that they can be fitted inside any household or decorative items such as toys, clocks or picture frames. My husband and I accustomed to have bicycles until these were stolen from my garage. Having a CCTV camera installed in the home, office or business allows you to observe the location from anywhere inside world via computer or closed circuit television. Gone then would be the days of aimlessly scrolling though lines of text and website URL's.

However, if you plan to utilize it commercially then you are able to go for special equipments that may let you connect hundreds and even thousands of televisions. Establishing a synopsis via placements of cameras within key areas in the office environment or retail store is an integral part of CCTV security systems.

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The obvious question is how you will be handling the storage in the data. Simply visit there Web site ( to view there most present-day items. As well as within the event you intend to proceed for far more progress safety methods, then you might go while using remotely reviewed cameras acquiring cellular device checking. You are busy going from work to getting the kids to after school activities. For prosecution, there isn't any better evidence than a time stamped video to provide in court.

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