Uncountable Excluded Point Space is not Second-Countable/Proof 1
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Theorem
Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space.
Then $T$ is not second-countable.
Proof
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.
$\blacksquare$