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.

$\blacksquare$

Also see

• Uncountable Discrete Space is not Second-Countable

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $2$. Countable Discrete Topology: $8$