# Subset of Indiscrete Space is Everywhere Dense

## 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.

$\blacksquare$