Uniform structure on subspace

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.