# Question from Introduction to Topology by Mendelson

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.

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Don't over-complicate matters! You have it. If $A\subset O$ then $A\cap O = A$. Remember that you're starting with the assumption that $A$ is a subset of $O$.