general topology on real line

Question

Given open sets A and B on Real line. Construct example such that $A\cap \overline B$, $\overline A \cap B$, $\overline {A\cap B}$, $\overline A \cap \overline B$ are distinct. Also give an example of two intervals such that $A\cap \overline B \not \subset \overline {A\cap B}$.

Answer 1

I think I found something, but I'm not quite sure whether it is a little bit too complicated: \begin{equation*} A= \bigcup_{n\in\mathbb{N}} \bigg]\frac{1}{n+1}, \frac{1}{n}\bigg[, \quad B=\bigcup_{n\in\mathbb{N}}\bigg]\frac{2n+3}{(n+1)(n+2)},\frac{2n+1}{n(n+1)}\bigg[. \end{equation*} Try it with these sets.

Answer 2

Let $A=(0,2)\cup (3,4)$ and $B=(1,3).$

$A\cap \overline B=((0,2)\cup (3,4))\cap [1,3]=[1,2).$

$\overline A\cap B=([0,2]\cup [3,4])\cap (1,3) =(1,2].$

$\overline A\cap \overline B=([0,2]\cup [3,4])\cap [1,3]=[1,2]\cup \{3\}.$

$\overline {A\cap B}=\overline {(1,2)}=[1,2].$

Let $A'=[0,1]$ and $B'=(1,2].$

$1\in [0,1]\cap [1,2]=A'\cap\overline {B'}.$

$1\not \in \emptyset =A'\cap B'=\overline {A'\cap B'}.$