Countable dense homogeneity and the Cantor set

It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says that for any crowded Hausdorff space $X$ of countable $π$-weight, if ${}^ω{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set.

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