# Dense-in-itself Subset of T1 Space is Infinite

## Theorem

Let $T = \struct {S, \tau_p}$ be a topological space which is $T_1$ (Fréchet).

Let $H \subseteq T$ be dense-in-itself.

Then $H$ is infinite.

## Proof

Aiming for a contradiction, suppose $H$ is finite.

From Finite $T_1$ Space is Discrete, $H$ has the discrete topology.

From Discrete Space is not Dense-In-Itself it then follows that $H$ can not be dense-in-itself.

So for $H$ to be dense-in-itself, it must be infinite.

$\blacksquare$