# Definition:Closure (Topology)/Definition 1

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## Contents

## Definition

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

Let $H \subseteq S$.

The **closure of $H$ (in $T$)** is defined as:

- $H^- := H \cup H'$

where $H'$ is the derived set of $H$.

That is, $H^-$ is the union of $H$ and its limit points.

## Notation

The topological closure of $H$ is variously denoted:

- $\map \cl H$
- $\map {\mathrm {Cl} } H$
- $\overline H$
- $H^-$

Of these, it can be argued that $\overline H$ has more problems with ambiguity than the others, as it is also frequently used for the set complement.

$\map \cl H$ and $\map {\mathrm {Cl} } H$ are cumbersome, but they have the advantage of being clear.

$H^-$ is neat and compact, but has the disadvantage of being relatively obscure.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, $H^-$ is notation of choice, although $\map \cl H$ can also be found in places.

## Also see

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

## Sources

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (previous) ... (next): $2.26$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.7$: Definitions: Definition $3.7.11$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $9.$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**closure**