A zero-dimensional metrizable CDH space X is constructed such that X^2 is not CDH, because X^2 has exactly continuum many pairwise non-equivalent countable dense subsets.
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Zero-dimensional metrizable CDH space $X$ such that $X^2$ is not CDH
A zero-dimensional metrizable CDH space X is constructed such that X^2 is not CDH, because X^2 has exactly continuum many pairwise non-equivalent countable dense subsets.