Is an open subset of an open set of a topological space open?

Question

If $X$ is an topological space and $ U \subseteq V \subseteq X$ with $V$ open in $X$ and $U$ open in $V$, is it true that $U$ is open in $X$?

Answer 1

Yes. $U$ open in $V$ means that $U=V\cap W$ for some $W$ open in $X$. So $U$ is an intersection of two open sets in $X$.

Answer 2

Hint: $U $ is open in $V $ iff $U=V\cap S $ for some $S $ open in $X $.