Is there a standard name for set $M$ for which $M = \overline{M^0}$?
$M^0$ is interior and $\overline{M}$ is closure.
Often I work with well behaved sets and functions. I have in mind continuous function defined open set and this function can be extended to continuous function on closure of this set. I find this too long to say, I would like to say I have continuous function on "name for $M$" ($M = \overline{M^0}$).
How "wild" such sets can be? I would like to get some examples of wild sets satisfying $M = \overline{M^0}$, in particular such $M$ that $\partial M$ is not manifold (for example two kissing balls).
This is called a "regular closed set". Older texts called them closed domains. The dual notions are regular open sets ($O = \operatorname{Int} \overline{O}$) or open domains.