Does there always exist an open set in bigger space that lying over a given open set in smaller space?

Question

Let $X$ be a metric or topological space. $Y\subseteq X$ be its subspace. Let $V\subseteq Y$ be open with respect to the smaller subspace $Y$ (i.e. $V$ may be not open with respect to $X$). Then I wonder is there always an open set $U$ with respect to $X$, such that $U\cap Y=V$?

Answer 1

Yes, there is. For each $x\in V$, there is some $r_x>0$ such that $D(x,r_x)\cap Y\subset V$, by definition of open set in a metric space. So, take $U=\bigcup_{x\in V}D(x,r_x)$, and that will be an open subset of $X$ such that $U\cap Y=V$.

Answer 2

This is simply the definition of the subspace topology. A subset in the subspace is open if and only if it is the intersection of an open subset in the larger space with the subspace.