Boundary

Every subset of $S$ of a topological space $(X,\tau )$ divides it into three different regions such that $X$ is always equal to the union of those regions.

Interior

Definition: Interior of a Set

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 $N(p)$ contained in $S$. The interior of $S$ is the set of all of its interior points.

$$\{p\in X\mid \exists N(p):N(p)\subseteq S\}$$

NOTATION

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

Characterizations

Theorem: Interior via Open Sets

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

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

$$\mathrm{int}S=\bigcup \{I\subseteq S\mid I\in \tau \}$$

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.

Boundary

Definition: Boundary

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

The boundary of $S$ is the set of all points $x\in X$ such that every neighbourhood of $x$ contains at least one point of $S$ and at least one point of its Complement $X\setminus S$.

NOTATION

$$\partial S{\partial }_{X}S{\mathrm{Bd}}_{X}S$$

Definition: Boundary Point

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

A point $p\in X$ is a boundary point of $S$ iff it belongs to the Boundary of $S$.

$$p\in \partial S$$

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 exterior $\mathrm{Ext}S$ 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.