Cantor Space is not Scattered
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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: $3$