I would like to rigorously show that $(\overline{X^\mathsf{c}})^\mathsf{c} = X^{\circ}$, that is, the complement of the closure of the complement of X equals the interior of X. 

I am TAing a class and today I gave the following proof to my students:

$\overline{X}= X^\circ \cup B$, where $X^\circ$ is the interior of X and $B$ is the boundary of X. 

Then, I just substitute in $X^\circ$ for $X$ to get:

$\overline{X^\mathsf{c}}= (X^\mathsf{c})^\circ \cup B$.

Then, by demorgan laws: $(\overline{X^\mathsf{c}})^\mathsf{c}=((X^\mathsf{c})^\circ)^\mathsf{c} \cap B^\mathsf{c}$.

However, this is a lot of work and would require additional proof for each step. For example, while it is intuitive that $((X^\mathsf{c})^\circ)^\mathsf{c}=X^\circ$, it takes quite a lot of effort to prove. 

Would anyone know of any magic bullet style proof that is concise and doesnt rely on the justification of multiple steps? Thanks!