Proximity 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 proximity space
View other induced structures on subspaces

Definition

Suppose $(X,\delta _{X})$ is a proximity space, i.e., $X$ is a set and $\delta _{X}$ is a proximity structure on $X$. Suppose $Y\subseteq X$. The induced proximity structure on $Y$, denoted $\delta _{Y}$, is defined as follows:

$\ \forall \ A,B\subseteq Y,\ A\ \delta _{Y}\ B\iff A\ \delta _{X}\ B$.