Is it true that $\overline{A \cap \overline{B}} = \overline{A \cap {B}}$?

Question

I was trying to understand Willard's proof (the book is called Topology) about local compacteness that any subset of a Hausdorff space is the intersection of an open set and a closed set. Then I came across this post: Locally compact subspace is an intersection of an open and closed set. But I can't prove $$(1) \quad \overline{W_x \cap \overline{M}} = \overline{W_x \cap {M}}$$ (wich is very similar to what Willard does $\overline{W_x \cap {M}} \subseteq X \implies \overline{W_x} \cap M \subseteq X$).

How does one prove $(1)$? Thank you for the help.

you can't prove this, because it's false - it could be that $A \cap B$ is empty and $A \cap \bar B$ is not — Matthew Towers, Nov 02 '22 at 18:42
I found out exactly what I was looking for in @QuantumSpace 's answer in here: https://math.stackexchange.com/questions/3584098/a-locally-compact-and-dense-subset-of-a-hausdorff-space-is-open — Diogo Santos, Nov 02 '22 at 22:30

user140776 · Accepted Answer · 2022-11-02T19:32:43.337

4

No. Let $A=(0,1)\cup \{2\}$, and $B=(2,3)$ then $\overline{A\cap \overline{B}}=\{2\}\ne\emptyset =\overline{A\cap B}$,

edited Nov 02 '22 at 19:32

answered Nov 02 '22 at 19:03

user140776

1,780

@Angelo yes. Thanks! – user140776 Nov 02 '22 at 19:33
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You could also have let $A=[0,2]$ and $B=(2,3)$. – Angelo Nov 02 '22 at 19:33

Is it true that $\overline{A \cap \overline{B}} = \overline{A \cap {B}}$?

1 Answers1