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A topological space $X$ is called totally disconnected if for any distinct points $x,y\in X$ there exists a clopen set $U\subseteq X$ such that $x\in U$ and $y\notin U$.

In 1973 Roman Pol proved that every separable metrizable space $M$ containing a topological copy of any totally disconnected separable metrizable space, contains a topological copy of the Hilbert cube. This result of Pol shows that there is no universal totally disconnected separable metrizable spaces. On the other hand, the Cantor cube is a universal totally disconnected compact metrizable space.

Problem. Is there a totally disconnected Polish space containing a topological copy of each totally disconnected Polish space?

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  • $\begingroup$ Is a countable product of countable discrete sets a possible candidates, or do you know it doesn't work? $\endgroup$
    – YCor
    Commented Jul 22, 2022 at 12:24
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    $\begingroup$ @YCor The countable product of discrete spaces is zero-dimensional. On the other hand, there exist totally disconnected Polish space of arbitrarily high dimension, including strongly infinite-dimensional, see e.g. topology.nipissingu.ca/tp/reprints/v07/tp07113.pdf $\endgroup$
    – Taras Banakh
    Commented Jul 22, 2022 at 14:19

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