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:

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