Let
be a Hausdorff space, let
then there exists open sets
such that:
for all
.
for all
For
the Theorem is trivially true. Let
For
this is just the definition of Hausdorff spaces.
Next show that if the Theorem is true
for then it holds for
.
Choose open sets
such that
for all
.
for all
Next, for each
choose
and
such
that
and
for
all
for for all
Define
and
Note that
for all
.
for all
since
and
for all
for all
since