Subset of Indiscrete Space is Everywhere Dense

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Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Let $H \subseteq S$ such that $H \ne \varnothing$.


Then $H$ is everywhere dense.


Proof

From Limit Points of Indiscrete Space, every point of $T$ is a limit point of $H$.

Hence $H$ is everywhere dense by definition.

$\blacksquare$


Sources