## A Totally Disconnected Topology for the Real Numbers

Theorem

with the topology consisting of all half open, half closed intervals of the formis totally disconnected.

Proof

A topological spaceis said to be totally disconnected if for eachopen nonempty setsexist such thatand

Takewith

Define

andare disjoint, open and nonempty. and

Alsosowith the topology defined above is totally disconnected.