Uniform structure on subspace: Difference between revisions

Latest revision as of 01:43, 25 November 2008

This article describes the induced structure on any subset (subspace) corresponding to a particular structure on a set: the structure of a uniform space
View other induced structures on subspaces

Definition

Definition in terms of entourages

Suppose $(X,{\mathcal {U}})$ is a uniform space: $X$ is a set and ${\mathcal {U}}$ is a uniform structure on $X$ . Suppose $Y\subseteq X$ . The induced uniform structure on $Y$ , denoted ${\mathcal {U}}_{Y}$ , is defined as follows:

${\mathcal {U}}_{Y}=\{U\subseteq Y\times Y\mid \exists \ V\in {\mathcal {U}},\ U=V\cap (Y\times Y)\}$ .

Definition in terms of coarsest uniform structures

Suppose $(X,{\mathcal {U}})$ is a uniform space and $Y\subseteq X$ . The induced uniform structure on $Y$ is the coarsest uniform structure on $Y$ for which the inclusion map from $Y$ to $X$ is a uniformly continuous map.

@@ Line 1: / Line 1: @@
 {{induced subspace structure|uniform space}}
-==Statement==
+==Definition==
+===Definition in terms of entourages===
 Suppose <math>(X,\mathcal{U})</math> is a [[uniform space]]: <math>X</math> is a set and <math>\mathcal{U}</math> is a uniform structure on <math>X</math>. Suppose <math>Y \subseteq X</math>. The induced uniform structure on <math>Y</math>, denoted <math>\mathcal{U}_Y</math>, is defined as follows:
 <math>\mathcal{U}_Y = \{ U \subseteq Y \times Y \mid \exists \ V \in \mathcal{U}, \ U = V \cap (Y \times Y) \}</math>.
+===Definition in terms of coarsest uniform structures===
+Suppose <math>(X,\mathcal{U})</math> is a [[uniform space]] and <math>Y \subseteq X</math>. The induced uniform structure on <math>Y</math> is the [[defining ingredient::coarser uniform structure|coarsest uniform structure]] on <math>Y</math> for which the inclusion map from <math>Y</math> to <math>X</math> is a [[defining ingredient::uniformly continuous map]].