Countable Excluded Point Space is Second-Countable/Proof 2

Theorem

Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space.

Then $T$ is a second-countable space.

Proof

We have:

Countable Discrete Space is Second-Countable

Excluded Point Topology is Open Extension Topology of Discrete Topology

The result follows from Condition for Open Extension Space to be Second-Countable.

$\blacksquare$