Cantor Space is Second Category in Itself
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Theorem
Let $\left({\mathcal C, \tau_d}\right)$ be the Cantor set considered as a topological subspace of the real number space $\R$ under the Euclidean topology $\tau_d$.
Then $\mathcal C$ is second category (non-meager) in itself.
Proof
We have that the Cantor set is a complete metric space.
Second category should now follow directly, but that is a result still needing to be proved.
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 29: \ 4$
