Definition:Clopen

Definition

Let $T$ be a topological space.

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

Then $S$ is described as clopen.

From Open and Closed Sets in a Topological Space, we have that in any topological space $T$, both $T$ and $\varnothing$ are clopen in $T$.

Linguistic Note

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

Also known as

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

Also see

• Results about clopen sets can be found here.

Sources

• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): $\S 1.1$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$