Cantor Space is Dense-in-itself
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Theorem
Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is dense-in-itself.
Proof
Let $U \in \tau_d$ be open in $T$.
Let $p \in U$.
Then $\exists x \in U: \exists \epsilon \in \R: \map d {x, p} < \epsilon$.
Hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set: $3$