Discrete Space is not Dense-In-Itself

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Theorem

Let $T = \struct {S, \tau}$ be a discrete topological space.

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


Proof

By definition, $T$ is dense-in-itself if and only if it contains no isolated points.

The result follows from Topological Space is Discrete iff All Points are Isolated.

$\blacksquare$


Sources