# Countable dense homogeneity and the Cantor set

## TL;DR

The paper demonstrates that under the Continuum Hypothesis, there exists a compact Hausdorff space that is countable dense homogeneous but does not contain a Cantor set, contrasting previous results.

## Contribution

It constructs a specific example of a compact Hausdorff space with these properties, showing a new independence result related to the Cantor set and countable dense homogeneity.

## Key findings

- Existence of a compact Hausdorff space under CH with specified properties
- Contrasts with previous results linking countable dense homogeneity and the Cantor set
- Highlights the role of set-theoretic assumptions in topology

## Abstract

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 $\pi$-weight, if ${}^\omega{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set.