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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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1 Answer 1

up vote 3 down vote accepted

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 $O$.

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O man, thank you. I completely over looked that in the problem statement. It's my first time using this site and I think I need to give credit or something like that? How do I go about doing that? –  Shant Danielian Jul 20 '13 at 4:19
    
You can upvote any question or any answer. When you've asked the question, you can accept an answer you like, as well. It's good to do so, because lots of readers look for questions that are still open. –  Ted Shifrin Jul 20 '13 at 4:24
    
Great, I guess I'll go ahead and upvote this and accept it. Thanks again for the help. –  Shant Danielian Jul 20 '13 at 4:27

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