# Definition:Boundary (Topology)

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

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.

Then 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 $H$.

## Also known as

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

## Notation

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

- $\partial H$
- $\map {\operatorname b} H$
- $\map {\operatorname {Bd} } H$
- $\map {\operatorname {fr} } H$ (where $\operatorname {fr}$ 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$: Closures and Interiors - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.7$: Definitions: Definition $3.7.31$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**boundary**