Uncountable Excluded Point Space is not Second-Countable

Theorem
Let $T = \left({S, \tau_{\bar p}}\right)$ be an uncountable excluded point space.

Then $T$ is not second-countable.

Proof 1
Let $H = S \setminus \left\{{p}\right\}$ where $\setminus$ denotes set difference.

By definition, $H$ is an uncountable discrete space.

The result follows from Uncountable Discrete Space is Not Second-Countable.

Proof 2
We have:


 * Uncountable Discrete Space is Not 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