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.

FACT 1. Disjoint closed subsets of #92;mathbb#123;N#125;#92;mathbb#123;N#125; may be separated by disjoint clopen sets.

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.

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