Theorem
The set of rational numbers
with the Euclidean topology is totally disconnected.
Proof
Let
and let
An irrational number
exists satisfying
Define sets
and
by
and
sois totally disconnected.
Theorem
The set of rational numberswith the Euclidean topology is totally disconnected.
Proof
Letand let
An irrational numberexists satisfying
Define setsandby
andsois totally disconnected.