clopen en · ADJ
Pronunciation
- /ˈkləʉ̯.pən/ (Australia)
- /ˈkləʊ̯.pən/ (Received-Pronunciation)
- /ˈklo.pən/ (Scotland)
- klō′pən
- /ˈkloʊ̯.pən/ (Canada, General-American)
- /ˈkloː.pen/ (India)
- /ˈklɐʉ̯.pən/ (Australia, New-Zealand)
Etymology
Etymology tree English closed Proto-Indo-European *upó Proto-Germanic *ub Proto-Germanic *upp Proto-Germanic *upanaz Proto-West Germanic *opan Old English open Middle English open English open blend English clopen Blend of closed + open.
Meanings
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(not-comparable, of a set in a topological space) Both open and closed.
Theorem 3. 1) Every two disjoint closed sets in a totally disconnected compact space #92;mathfrak#123;Q#125; 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.
2002, D. A. Vladimirov, Boolean Algebras in Analysis, page 133:FACT 1. Disjoint closed subsets of #92;mathbb#123;N#125;#92;mathbb#123;N#125; may be separated by disjoint clopen sets.
1999, S. J. Dilworth, “On the extensibility of certain homeomorphisms and linear isometries”, in Krzysztof Jarosz, editor, Function Spaces: Proceedings of the Third Conference on Function Spaces (Contemporary Mathematics), volume 232, American Mathematical Society, page 124:A subset of a metric space is clopen iff it is both closed and open. A metric space is called zero-dimensional iff there is a base for the open sets consisting of clopen sets.
1990, Gerald A Edgar, Measure, Topology, and Fractal Geometry, page 80:
Relateds
the half-open interval defined there is not to be confused with the clopen set