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:

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

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.