# Definition:Boundary (Topology)

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

## Contents

## 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 $T$:

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

## Also known as

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

## Also defined as

Some sources define the boundary of a subset 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 $H^b$.

## Also see

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

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**boundary**(of a surface)