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 ℕN may be separated by disjoint clopen sets.

Theorem 3. 1) Every two disjoint closed sets in a totally disconnected compact space 𝔔 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.

James is on vacation, so I've got to do a clopen this weekend to cover his shifts.

Clopen

Meaning

Adj

Noun

Rhymes