Find a counter example

Question

The interior of the union is the union of the interiors. $\text{int}\left(A\cup B\right) = \text{int}(A) \cup \text{int}(B)$

I'm not too sure about to get started with this one. Any hints so as to help me to understand what I need to do?

Attempt

$A=[a,b)$

$B=[c,d)$

If I just do a normal union I get this:

$[a,b)\cup[c,d)=\{a,c\}$

If I take the interior union of the sets I get:

$\text{int} A=(a,b)$

$\text{int} B=(c,d)$

And so:

$\text{int} A \cup \text{int} B = \{0\}$

However, if I do the following:

$\text{int} (A \cup B)$

I get $(a,c)$

Answer 1

As you're asking for a hint, I suggest trying to find intervals $A$ and $B$ as counter examples.

More hints:

$A=[0,1), B = [1,2]$

Answer 2

One has $\text{int}A\subset A\subset A\cup B$ and $\text{int}B\subset B\subset A\cup B$ and so $\text{int}A\cup\text{int}B\subset A\cup B$. The interior of $A\cup B$ is the largest open contained in $A\cup B$ and $\text{int}A\cup\text{int}B$ is open as the Union of two interiors ie two open sets and so

$$\text{int}A\cup\text{int}B\subset\text{int}(A\cup B)$$

In $\Bbb{R}$ consider $A=[0,1)$ and $B=[1,2]$. We have $\text{int}(A\cup B)=(0,2)$ and $\text{int}A\cup\text{int}B=(0,1)\cup(1,2)$

Answer 3

Another nice counter example:

If singletons are closed sets in your space (which they usually are) and if they are not open sets (which, except for discreet topologies, they are), you can take

$$A=\{x_0\}\\ B=X\setminus \{x_0\}$$

in which case $B$ is open (so it is its own interior), and $A$ is not open (and has an empty interior), so

$$\text{int}(A)\cup \text{int}(B) = \emptyset\cup B=B\neq X =\text{int}(A\cup B)$$

Answer 4

If $D\subseteq X$ is dense and $D^c$ is dense as well then $\text{int}D=\varnothing=\text{int}D^c$ and $\text{int}D\cup D^c=\text{int}X=X$.

Example: $D=\mathbb Q\subseteq\mathbb R$.