Let $X$ be a metric space and $Y\subseteq X$.
For every $a\in X$, $a$ is an interior point of $Y$ if there is $\eps>0$ such that $\mathcal{B}_\eps(a)\subseteq Y$.
The set of all interior points of $Y$ is called the interior of $Y$:
\[\inter Y = \{a\in X : (\exists\eps>0 : \mathcal{B}_\eps(a)\subseteq Y)\}.\]