- Rhymes: -əʊpən
clopen (not comparable)
- (topology, of a set in a topological space) Both open and closed.
1990, Gerald A Edgar, Measure, Topology, and Fractal Geometry, page 80:
- 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.
- 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, 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.