Cantor Space is Non-Meager in Itself

Theorem

Let $T = \struct {\CC, \tau_d}$ be the Cantor space.


Then $T$ is non-meager in itself.


Proof

We have that the Cantor Space is Complete Metric Space.

By Baire Category Theorem, a complete metric space is also a Baire space.

The result then follows by Baire Space is Non-Meager.

$\blacksquare$


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: $4$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.