topology homework question so I got this question for homework:
let $x$ be a topological space and let $A \subset C$. one sets $\alpha(A) = \mathrm{Int}(\bar{A})$, and $\beta(A) = \overline{\mathrm{Int}(A)}$. Prove if $A$ is open, then $A \subset \alpha(A)$ and if $A$ is closed, then $\beta(A) \subset A$. 
deduce that for any $A \subset X$, $\alpha(\alpha(A))=\alpha(A)$ and $\beta(\beta(A))=\beta(A)$
any kind of help will be appreciated, thanks in advance! 
 A: Hint: if $A$ is open, $A=\text{Int}(A)$ and if $A$ is closed, $A=\overline{A}$. Also, when $A\subset B$, $\text{Int}(A)\subset\text{Int}(B)$ and $\overline{A} \subset \overline{B}$. Now use that $\text{Int}(A)\subset A\subset \overline{A}$.
A: Use the following (prove them if they are not clear to you):
1) for any subset:$A\subseteq \overline A$. 
2) if $U\subseteq A$ is open, then $U\subseteq \rm {Int}(A)$.
3) if $A$ is open, then $A\subseteq \overline A$ is an open set in $\overline A$. What do you conclude?
Can you tackle the rest?
A: If $A$ is open: $A \subset \overline{A}$, so $A \subset \operatorname{Int}(\overline{A}) = \alpha(A)$, as the interior of $\overline{A}$ is the largest open subset contained in a $\overline{A}$, and $A$ is one of these open sets.
If $B$ is closed: $\operatorname{Int}(B) \subset B$ and the closure of $\operatorname{Int}(A)$ is the smallest closed set containing it, and $B$ is a closed set containing it, so $\beta(B) = \overline{\operatorname{Int}(B)} \subset B$.
If $A \subset X$, then $\alpha(A)$ is open (it's an interior) so by the first part $\alpha(A) \subset \alpha(\alpha(A))$. On the other hand, $\alpha(A) = \operatorname{Int}(\overline{A}) \subset \overline{A}$, so $\overline{\alpha(A)} \subset \overline{\overline{A}} = \overline{A}$, and taking the interior on both sides of the last gives us $\alpha(\alpha(A)) = \operatorname{Int}(\overline{\alpha(A)}) \subset \operatorname{Int}(\overline{A}) = \alpha(A)$. So the required identity follows.
For $\beta(A)$, we have an analogous argument, where one inclusion follows from the second fact, and the other from standard facts.
