R with the discrete topology

Theorem

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

Then $T$ is first-countable.

Proof

From Point in Discrete Space is Neighborhood, every point $x \in S$ is contained in an open set $\set x$.

From the definition of local basis, it is clear that $\set {\set x}$ is [trivially] a local basis at $x$.

That is, that every open set of $S$ containing $x$ also contains at least one of the sets of $\set {\set x}$.

Equally trivially, we have that $\set {\set x}$ is countable.

Hence the result by definition of first-countable.

$\blacksquare$

Sources

• 1978:Lynn Arthur Steenand J. Arthur Seebach, Jr.: Counterexamples in Topology[2nd ed.]... [previous]... [next]: Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $7$

• Proven Results
• First-Countable Spaces
• Discrete Topology