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