clopen
Definition from Wiktionary, the free dictionary
English[edit]
Etymology[edit]
Pronunciation[edit]
 Rhymes: əʊpən
Adjective[edit]
clopen (not comparable)
 (topology, of a set in a topological space) Both open and closed.

1990, Gerald A Edgar, Measure, Topology, and Fractal Geometry^{[1]}, page 80:
 A subset of a metric space is clopen iff it is both closed and open. A metric space is called zerodimensional iff there is a base for the open sets consisting of clopen sets.
 1999, S. J. Dilworth, On the extensibility of certain homeomorphisms and linear isometries, Krzysztof Jarosz (editor), Function Spaces: Proceedings of the Third Conference on Function Spaces, American Mathematical Society, Contemporary Mathematics, Volume 232, page 124,
 FACT 1. Disjoint closed subsets of may be separated by disjoint clopen sets.

2002, D. A. Vladimirov, Boolean Algebras in Analysis^{[2]}, page 133:
 Theorem 3. 1) Every two disjoint closed sets in a totally disconnected compact space are separated by disjoint clopen sets.
 2) Every algebra of clopen sets which separates the points of a totally disconnected compact space contains all clopen sets.
