Countable Discrete Space is Second-Countable

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Let $S$ be a countable set, thereby making $\vartheta$ the countable discrete topology on $S$..

Then $T$ is second-countable.

If $S$ is uncountable, then $T$ is not second-countable.

Proof
From Basis for Discrete Topology, the set:
 * $\mathcal B := \left\{{\left\{{x}\right\}: x \in S}\right\}$

is a basis for $T$.

There is a trivial one-to-one correspondence $\phi: S \leftrightarrow \mathcal B$ between $S$ and $\mathcal B$:
 * $\forall x \in S: \phi \left({x}\right) = \left\{{x}\right\}$

Let $S$ be countable.

Then $\mathcal B$ is also countable by definition of countability.

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

Let $S$ be uncountable.

Then from Uncountable Discrete Space has No Countable Basis it follows that $T$ can not be second-countable.