Metrisability is hereditary property

Question

I am trying to show metrisability is topological property. If X is a topological space induced by d and $Y$ is a subspace, I am supposed to show that d restricted to Y x Y induce subspace topology on Y. If $B$ be a member of the subspace Topology I cannot show it is union of open balls generated by d restricted to Y x Y. Please help. I have done the other implication.

Answer 1

If $B$ is an open subset of $Y$, with respect to the subspace topology, then $B=O\cap Y$, for some open subset of $X$. For each $x\in B$, let $r>0$ be such that $B_r(x)\subset O$. Then $B_r(x)\cap Y\subset O\cap Y=B$. Therefore,$$B=\bigcup_{x\in B}B_r(x)\cap Y,$$and, on the other hand, $B_r(x)\cap Y$ is the open ball in $Y$ centered at $x$ and whose radius is $r$.