Subset of Excluded Point Space is not Dense-in-itself

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Theorem

Let $T = \struct {S, \tau_{\bar p} }$ be a excluded point space such that $S$ is not a singleton.

Let $H \subseteq S$.


Then $H$ is not dense-in-itself.


Proof

From Limit Points in Excluded Point Space, the only limit point of $H$ is $p$.

So by definition, all points of $H$ are isolated in $H$ except $p$.

So if $H \ne \set p$, $H$ contains at least one point which is isolated in $H$.

As for $p$ itself, from Singleton Point is Isolated we have that $p$ is itself isolated in $\set p$.

So if $H = \set p$, $H$ also contains one point which is isolated in $H$.

Hence the result, by definition of dense-in-itself.

$\blacksquare$


Sources

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