Interior

Definition

Let $X$ be a topological space and let $S \subseteq X$. The interior of $S$ is the union of all open sets $U$ of $X$ with $U \subseteq S$. It is written $\operatorname{int} S$ or $S^\circ$. A point of $\operatorname{int} S$ is an interior point of $S$.

Remarks

A point $x$ is an interior point of $S$ if and only if there is an open set $U$ with $x \in U \subseteq S$. In a metric space, $x$ is an interior point of $S$ if and only if $B(x, \varepsilon) \subseteq S$ for some $\varepsilon > 0$. In $\mathbb{R}$, the interior of $[0, 1]$ is $(0, 1)$, and the interior of $\mathbb{Q}$ is $\emptyset$.

Explanations

L1 human

The interior of a set is what remains when you remove its edge points. A point is in the interior when a whole small neighbourhood of it lies in the set.