Countable Excluded Point Space is Second-Countable/Proof 2

Theorem
Let $T = \left({S, \tau_{\bar p}}\right)$ 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