Cantor Space satisfies all Separation Axioms
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Theorem
Let $T = \struct {\mathcal C, \tau_d}$ be the Cantor space.
Then $T$ satisfies all the separation axioms.
Proof
We have that the Cantor space is a metric subspace of the real number space $\R$, and hence a metric space.
The result follows from Metric Space fulfils all Separation Axioms.
$\blacksquare$
Sources
- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 29: \ 2$