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Let $A$ and $B$ be subsets of $\mathbb{R}^n$. If $A$ is open, and $B$ is arbitrary, does one always have the following inclusion $A \cap \operatorname{cl}(B) \subseteq \operatorname{cl}(A\cap B)$.

Is there a counter-example? I can't seem to find a way to prove it.

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This question looks like homework. If it is, please use the [homework] tag. – Srivatsan Oct 24 '11 at 21:48
up vote 2 down vote accepted

The statement is true. The proof just uses the relevant definitions.

Suppose $x \in A \cap \operatorname{cl} (B)$, i.e., $x \in A$ and $x \in \operatorname{cl}(B)$. The latter condition means that if $U$ is an open set containing $x$, then $U$ intersects $B$ (nontrivially).

Now, pick any open set $V$ containing $x$. Then $V \cap A$ is an open set containing $x$. Then by the preceding paragraph, $(V \cap A) \cap B \neq \emptyset$, which is equivalent to $V \cap (A \cap B) \neq \emptyset$. This means that $x \in \operatorname{cl}(A \cap B)$.

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Beat me to it ;-) – Asaf Karagila Oct 24 '11 at 21:49
And me. :^) $ $ – Henning Makholm Oct 24 '11 at 21:50
@Henning I was just posting this comment under your answer, but you deleted it before I could submit it: "Wow, will I ever learn to write so compactly?" :). – Srivatsan Oct 24 '11 at 21:52
I spent some time tersifying it, because I didn't want it to deliver a too complete homework solution. – Henning Makholm Oct 24 '11 at 21:54
@Henning Pity I ended up doing just that (i.e., deliver complete solution). I guess I should be careful from now on. – Srivatsan Oct 24 '11 at 21:56

the closure is closed, and the intersection of a closed set and an open set is not always a closed set

a conter exemple may be: A=]0,2[ B=]1,3[

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Um, so does that actually answer the question? – Henning Makholm Oct 24 '11 at 21:51
$A=(0,2)$, $B=(1,3)$ is not a counterexample. We get $A\cap\mathrm{cl}(B)=[1,2)$ and $\mathrm{cl}(A\cap B)=[1,2]$. – Henning Makholm Oct 24 '11 at 21:57
i just noticed that its $A \cap \operatorname{cl}(B) \subseteq \operatorname{cl}(A\cap B)$ and not $A \cap \operatorname{cl}(B) = \operatorname{cl}(A\cap B)$ – Hassan Oct 24 '11 at 22:00

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