Definition:Clopen Set

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