Discrete Space is First-Countable

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 Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $7$
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