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 #92;mathbb#123;N#125;#92;mathbb#123;N#125; may be separated by disjoint clopen sets.1999, S. J. Dilworth, “On the extensibility of certain homeomorphisms and linear isometries”, in Krzysztof Jarosz, editor, Function Spaces: Proceedings of the Third Conference on Function Spaces (Contemporary Mathematics), volume 232, American Mathematical Society, page 124:
1 more example
Theorem 3. 1) Every two disjoint closed sets in a totally disconnected compact space #92;mathfrak#123;Q#125; 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:
Related terms
the half-open interval defined there is not to be confused with the clopen set
clopen
noun
Etymology Blend of closed + open.
1
A pair of work shifts in which a worker works a closing shift one day and then works an opening shift the following day.
James is on vacation, so I've got to do a clopen this weekend to cover his shifts.