Countable Discrete Space is Second-Countable

Theorem
Let $T = \struct {S, \tau}$ be a countable discrete topological space.

Then $T$ is second-countable.

Proof
From Basis for Discrete Topology, the set:
 * $\BB := \set {\set x: x \in S}$

is a basis for $T$.

There is a trivial one-to-one correspondence $\phi: S \leftrightarrow \BB$ between $S$ and $\BB$:
 * $\forall x \in S: \map \phi x = \set x$

Let $S$ be countable.

Then $\BB$ is also countable by definition of countability.

So we have that $T$ has a countable basis, and so is second-countable by definition.

Also see

 * Uncountable Discrete Space is not Second-Countable