Uniform structure induces topology: Difference between revisions

Latest revision as of 21:26, 24 November 2008

Statement

Given a Uniform space (?) $(X,{\mathcal {U}})$ (i.e., a set $X$ with a uniform structure ${\mathcal {U}}$ ) we define a topology on $X$ as follows (thus turning $X$ into a Topological space (?)): A subset $V\subseteq X$ is said to be open if, for every $x\in V$ , there exists $U\in {\mathcal {U}}$ such that whenever $(x,y)\in U$ , we have $y\in V$ .

Often, when we talk of a uniform structure on a topological space, we mean a uniform structure whose induced topology is the given topology.

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 ==Statement==
-Given a [[uniform space]] <math>(X,\mathcal{U})</math> (i.e., a set <math>X</math> with a uniform structure <math>\mathcal{U}</math>) we define a topology on <math>X</math> as follows. A subset <math>V \subseteq X</math> is said to be '''open''' if, for every <math>x \in V</math>, there exists <math>U \in \mathcal{U}</math> such that whenever <math>(x,y) \in U</math>, we have <math>y \in V</math>.
+Given a [[fact about::uniform space]] <math>(X,\mathcal{U})</math> (i.e., a set <math>X</math> with a uniform structure <math>\mathcal{U}</math>) we define a topology on <math>X</math> as follows (thus turning <math>X</math> into a [[fact about::topological space]]): A subset <math>V \subseteq X</math> is said to be '''open''' if, for every <math>x \in V</math>, there exists <math>U \in \mathcal{U}</math> such that whenever <math>(x,y) \in U</math>, we have <math>y \in V</math>.
 Often, when we talk of a uniform structure on a ''topological'' space, we mean a uniform structure whose induced topology is the given topology.