Proof That Every Separable Metric Space is Second Countable
Theorem
If
is a separable metric space it is second countable.
Proof
Letbe a separable metric space. Let A be a countable dense subset of
so
Let
be the set of all open balls with centres in
and with rational radii:
and
are countable sets, hence so is
Take
where
is open in
There is an open ball
such that
Sinceis dense in
we can find
such that
Let
be a rational number satisfying
Then
Since
andis countable,is a countable base for the given topology on
