Definition:Clopen Set

Definition

Let $T = \left({S, \tau}\right)$ 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 $T = \left({S, \tau}\right)$, both $S$ and $\varnothing$ 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): $\S 1.1$
• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{I}: \ \S 1$