Interior

Definition: Interior of a Set

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

The interior of a set $S\subseteq X$ is the union of all open sets contained in $S$.

$$\bigcup \{I\subseteq S\mid I\in \tau \}$$

NOTATION

$$\mathring{S}{S}^{\circ }\mathrm{int}S{\mathrm{int}}_{X}S$$

Definition: Interior Point

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

A point $x\in X$ is an interior point of $S$ iff $x$ belongs to the interior of $S$.

Theorem

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

A point $p\in X$ is an interior point of $S$ if and only if it has a neighbourhood contained in $S$.

PROOF

TODO

Properties

Theorem: Interior is a Subset

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

The interior $\mathrm{Int}S$ of each subset $S\subseteq X$ is a subset of $S$.

$$\mathrm{Int}S\subseteq S$$

PROOF

Suppose that $\mathrm{int}S$ is not a subset of $S$. Then there must exist some $s\in \mathrm{int}S$ such that $s\notin S$. Since $s\in \mathrm{int}S$, there must exist some open set $O$ such that $s\in O$ and $O\subseteq S$. However, $s\notin S$ implies that $O$ is not a subset of $S$, which is a contradiction.