EDB — 0PR

[0PR] Let $E\subseteq X$, then $E$ is a metric space with the restricted distance $\tilde{d}=d{|}_{E\times E}$.

Show that $A\subseteq E$ is open in $(E,\tilde{d})$ (as defined at the beginning of this section) if and only there exists a set $V\subseteq X$ open in $(X,d)$ such that $V\cap E=A$.

(The second way of defining ”open” is used in topology.)