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 a neologism which, in general, has no meaning outside the specialized language of topology.
However, in societies which the retail industry has a reputation for exploiting its workers, clopen is used in the context of the same person performing a closing shift followed by the opening shift the following day.
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): clopen
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): clopen