English

Adjective

clopen ‎(not comparable)

1. (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.