# Definition:Clopen Set

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## Definition

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

Let $X \subseteq S$ such that $X$ is both open in $T$ and closed in $T$.

Then $X$ is described as **clopen**.

## Also known as

Earlier sources refer to **clopen sets** as **closed-open sets** or **open-closed sets**.

## Also see

- Open and Closed Sets in Topological Space: in any topological space $T = \struct {S, \tau}$, both $S$ and $\O$ are
**clopen**in $T$.

- Results about
**clopen sets**can be found here.

## Linguistic Note

The term **clopen set** is an obvious neologism which has no meaning outside the specialized language of topology.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets - 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 - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**clopen** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**clopen**