1. ## Topology exercise.

Picture attached.

2. ## Re: Topology exercise.

Originally Posted by vercammen
Picture attached.
You have posted an attachment without any comment whatsoever.
Do you expect someone to do the problem(s) for you?

3. ## Re: Topology exercise.

I expect any kind of help. Even hint or the first step.
Anyway, here is what I got

1. Clearly $\displaystyle \supset$ is always true, even when $\displaystyle G$ is not open. Let $\displaystyle x\in \overline{G\cap \overline A}$, and $\displaystyle O$ an open neighborh. of $\displaystyle x$. It necessarily meets a point of $\displaystyle G\cap \overline A$, say $\displaystyle y$. As $\displaystyle y\in\overline A$, and $\displaystyle G$ is a neighborhood of $\displaystyle y$, $\displaystyle G\cap A\cap O\neq \emptyset$. So $\displaystyle x\in \overline{G\cap A}$.

2. We consider the real line with usual topology, $\displaystyle A:=(0,1)$ and $\displaystyle G:=\{0\}$. The LHS is $\displaystyle \{0\}$ but the RHS is empty.

4. ## Re: Topology exercise.

What are some good books for a layperson who wants to teach himself set theory?

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