# Definition:Boundary (Topology)

*This page is about Boundary in the context of topology. For other uses, see Boundary.*

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

### Definition from Closure and Interior

The **boundary of $H$** consists of all the points in the closure of $H$ which are not in the interior of $H$.

Thus, the **boundary of $H$** is defined as:

- $\partial H := H^- \setminus H^\circ$

where $H^-$ denotes the closure and $H^\circ$ the interior of $H$.

### Definition from Neighborhood

$x \in S$ is a **boundary point** of $H$ if every neighborhood $N$ of $x$ satisfies:

- $H \cap N \ne \O$

and

- $\overline H \cap N \ne \O$

where $\overline H$ is the complement of $H$ in $S$.

The **boundary of $H$** consists of all the **boundary points** of $H$.

### Definition from Intersection of Closure with Closure of Complement

The **boundary of $H$** is the intersection of the closure of $H$ with the closure of the complement of $H$ in $T$:

- $\partial H = H^- \cap \paren {\overline H}^-$

### Definition from Closure and Exterior

The **boundary of $H$** consists of all the points in $H$ which are not in either the interior or exterior of $H$.

Thus, the **boundary of $H$** is defined as:

- $\partial H := H \setminus \paren {H^\circ \cup H^e}$

where:

## Also known as

The **boundary** of a subset $H$ of a topological space $T$ is also seen referred to as the **frontier of $H$**.

## Also defined as

Some sources define the boundary of a subset of a topological space $T = \struct {S, \tau}$ as:

However, this definition is of little use without rigorous definitions of **approach**, **inside** and **outside**.

Consequently, such imprecise definitions are not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Notation

The **boundary** of $H$ is variously denoted (with or without the brackets):

- $\partial H$
- $\map {\mathrm b} H$
- $\map {\mathrm {Bd} } H$
- $\map {\mathrm {fr} } H$ or $\map {\mathrm {Fr} } H$ (where $\mathrm {fr}$ stands for
**frontier**) - $H^b$

The notations of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$ are $\partial H$ and $H^b$.

## Examples

### Half-Open Real Interval

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\hointl a b$ be a half-open interval of $\R$.

Then the boundary of $\hointl a b$ is the set of its endpoints $\set {a, b}$.

### Open Unit Interval

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\openint 0 1$ be the open unit interval in $\R$.

Then the boundary of $\openint 0 1$ is the set of its endpoints $\set {0, 1}$.

### $\Z$ in $\R$

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\Z$ be the set of integers.

Then the boundary of $\Z$ in $\struct {\R, \tau_d}$ is $\Z$ itself.

### Reciprocals in $\R$

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $S$ be the set defined as:

- $S = \set {\dfrac 1 n: n \in \Z_{>0} }$

Then the boundary of $S$ in $\struct {\R, \tau_d}$ is $S \cup \set 0$.

### Rationals in Closed Unit Interval in $\R$

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $S$ be the set defined as:

- $S = \Q \cap \closedint 0 1$

where:

- $\Q$ denotes the set of rational numbers
- $\closedint 0 1$ denotes the closed unit interval.

Then the boundary of $S$ in $\struct {\R, \tau_d}$ is $\closedint 0 1$.

## Also see

- Results about
**set boundaries**can be found**here**.