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): $\text{I}: \ \S 1$