Countable Discrete Space is Second-Countable
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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
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: $8$