Exterior

Definition: Exterior of a Set

Let $(X,{\tau }_X)$ be a topological space and let $S$ be a subset of $X$.

The exterior of $S$ is the complement of its closure in $X$.

$$X\setminus \overline{S}$$

NOTATION

$$\mathrm{ext}S\mathrm{Ext}S$$

Definition: Exterior Point

Let $(X,{\tau }_X)$ be a topological space and let $S$ be a subset of $X$.

A point $p\in X$ is an exterior point of $S$ iff it belongs to the exterior of $S$.

Properties

Theorem: Exterior is Closed

Let $(X,\tau )$ be a topological space.

The $\mathrm{Ext}S$ of each subset $S\subseteq X$ is closed.

PROOF

The exterior of $S$ is the complement of its closure and since the closure is a closed set, the exterior is open.