CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
For $S \subset X$ a subset of a topological space $X$, a, interior point of $S$ is a point $x \in S$ which has a neighbourhood in $X$ that is contained in $S$. The union of all interior points is the interior $S^\circ$ of $S$.
In general, we have $S^\circ \subseteq S$. $S$ is open if and only if $S^\circ = S$.
Compare the topological closure $\bar{S}$ and frontier $\partial S = \bar{S} \setminus S^\circ$.