Suppose we have a topological space $X$, an element $x\in X$, and an open neighborhood $x \in O \subset X$. Further, suppose that $X$ is a topological subspace of $Y$.
I am trying to figure out under what conditions there must exist an open neighborhood of $x$ in $Y$?
ok, so the question as written is trivial since $Y$ is open.
The requirement $O = U \cap X$ for some $U \subset Y$ open is also true by definition of relative topology.
Thanks.