nLab
interior

Definition

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 $\overline{S}$ and frontier $\partial S=\overline{S}\setminus S^{\circ }$.