Theorem
A totally disconnected space
is Hausdorff.
Proof
Letbe totally disconnected and let
Since
is totally disconnected, disjoint open sets
exist such that
and 
Henceis Hausdorff.
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Theorem
A totally disconnected spaceis Hausdorff.
Proof
Letbe totally disconnected and let
Sinceis totally disconnected, disjoint open setsexist such thatand
Henceis Hausdorff.