Interior of a set

Definition · applies to sets · also called $\operatorname{int} A$, $A^{\circ}$

Elementary

The interior of a shape is the part on its inside, not including its edge.

Undergraduate (subsets of $\reals^n$)

A point $x$ is in the interior of $A$ if there exists an open ball around $x$ that is contained in $A$.

Graduate (topological spaces)

The interior of $A$ is the union of all open sets contained in $A$: the largest open subset of $A$.

Examples

The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$ : $\operatorname{int} D = \setdef{(x, y) \suchthat x^2 + y^2 < 1}$ High school
The interior of the closed unit disk $D$ is the open disk.
$[0, 1]$ : $\operatorname{int} [0, 1] = (0, 1)$ High school
The interior of $[0, 1]$ is $(0, 1)$, because no interval around $0$ or $1$ stays inside $[0, 1]$.
$[0, 1)$, as a subset of $\reals$ : $\operatorname{int} [0, 1) = (0, 1)$ High school
The interior of $[0, 1)$ is $(0, 1)$, because every point of $(0, 1)$ has a small interval around it within $[0, 1)$.
$(0, 1)$ : $\operatorname{int} (0, 1) = (0, 1)$ High school
The interior of $(0, 1)$ is $(0, 1)$, because every point of $(0, 1)$ has a small interval around it inside $(0, 1)$; that is, $(0, 1)$ is open.
$\mathbb{Q}$, as a subset of $\reals$ : $\operatorname{int} \mathbb{Q} = \emptyset$ Undergraduate
Every interval contains rational and irrational numbers, so $\mathbb{Q}$ has an empty interior and an empty exterior.

A set is open if and only if it equals its interior.
The interior, boundary, and exterior of a set are disjoint, and together they make up the whole space.