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.
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Theorem
The set of rational numberswith the Euclidean topology is totally disconnected.
Proof
Letand let
An irrational numberexists satisfying
Define setsandby
andsois totally disconnected.