Is it true that the intersection of the closures of sets $A$ and $ B$ is equal to the closure of their intersection?

Question

Is it true that the intersection of the closures of sets $A$ and $B$ is equal to the closure of their intersection? $ cl(A)\cap{cl(B)}=cl(A\cap{B})$ ?

Answer 1

No. Take $A=(-1,0), B=(0,1)$. Note that $0$ is in the closure of both of these sets.

Answer 2

No, it's not true. Look at this page, or this question.

A simple counterexample stolen from the question cited above is as follows:

Take $A = (0,1)$ and $B = (1,2)$. Then we have $$\operatorname{cl}(A) \cap \operatorname{cl}(B) = [0,1] \cap [1,2] = \lbrace 1 \rbrace $$ but $$\operatorname{cl}(A \cap B) = \operatorname{cl}(\emptyset) = \emptyset$$

Answer 3

No, the rationals and the irrationals (in the reals) are disjoint so $\operatorname{cl}(A \cap B) = \operatorname{cl}{\emptyset}=\emptyset$ while $\operatorname{cl}(A) = \operatorname{cl}(B) = \mathbb{R}$