# clopen

## 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 zero-dimensional iff there is a base for the open sets consisting of**clopen**sets.

- A subset of a metric space is
**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.

- FACT 1. Disjoint closed subsets of may be separated by disjoint
**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.