Identities involving the boundary, closure and interior of a set.

Let $(M,tau )$ be some topological space, and $A$ be a set in $M$.
I will denote $A^{circ}$ as the interior of $A$, $overline{A}$ the closure of $A$ and $partial A$ the boundary of $A$.

I am trying to collect as many identities involving the boundary, interior and closure of a set.

The ones that I have so far are $$partial A=partial A^c$$
$$partial A =overline{A} cap overline{(A^c)}$$
$$partial A=partial A^c$$
$$bar{A}= partial A cup A^{circ}$$
$$M=A^{circ} cup partial A cup (A^c)^{circ}$$
$$partial A= overline{A} – A^{circ}$$
Can you show some more identities like those along with their proofs?