I'm self studying Intro to Topology by Mendelson(3rd ed.) right now and I'm stuck on a book problem. In case anyone has the book handy, its problem 2 of chapter 3 section 6. The problem is as follows,
Let $O$ be an open subset of a topological space $X$. Prove that a subset $A$ of $O$ is relatively open in $O$ if and only if it is an open subset of $X$.
I was able to prove the forward direction by showing that $A$ is an intersection of open sets in $X$ and hence open. I'm having trouble proving the other direction.
I know that since $O$ is a subspace of $X$ all the open sets of $O$ take the form $O'\cap O$ for some open set $O'\subset X$. What I'm thinking is to show that $A=A\cap O$ or $A=O'\cap O$. The reason for the first is to show that implicitly $A\subset O$. I also realized that there is an open set $O''\supset O$ since $O=O''\cap O$.
Thanks for any help.