A Totally Disconnected Topology for the Real Numbers
Theorem
with the topology consisting of all half open, half closed intervals of the form
is totally disconnected.
Proof
A topological space
is said to be totally disconnected if for each
open nonempty sets
exist such that
and
Take
with
Define
and
are disjoint, open and nonempty. and
Also
sowith the topology defined above is totally disconnected.