Theorem
Every subspace
of a second countable space
is second countable.
Proof
Let
be a countable base for the second countable space
For any subspace
is a countable base for
Henceis second countable.
Also, any second countable space is separable. A space
contains a countable dense subset.
Letbe a second countable space with a countable base
For
select
The set
is a countable dense subset of
henceis separable.