Definition of "clopen"

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

FACT 1. Disjoint closed subsets of N N {\displaystyle \mathbb {N} ^{\mathbb {N} }} may be separated by disjoint 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

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

2002, D. A. Vladimirov, Boolean Algebras in Analysis‎, page 133