Cantor Space is not Scattered

Theorem

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


Then $T$ is not scattered.


Proof

By definition, $T$ is scattered if and only if it contains no non-empty subset which is dense-in-itself.

We have that Cantor Space is Dense-in-itself.

Hence the result by definition of a scattered space.

$\blacksquare$


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: $3$
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.