Countable Excluded Point Space is Second-Countable/Proof 1
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Theorem
Let $T = \struct {S, \tau_{\bar p} }$ be a countable excluded point space.
Then $T$ is a second-countable space.
Proof
Consider the set $\BB$ defined as:
- $\BB = \set {\set x: x \in S \setminus \set p} \cup \set S$
From Basis for Excluded Point Space, $\BB$ is a basis for $T$, and trivially has the same cardinality as $S$.
So by definition, if $S$ is countable, then $T$ is second-countable.
$\blacksquare$