Dense-in-itself Subset of T1 Space is Infinite
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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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $23 \text { - } 24$. Fort Space: $8$