Subset of Indiscrete Space is Everywhere Dense

Theorem
Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.

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

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.