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Let us recall a few definitions. A topological space is

  • metrisable if it is homeomorphic to a metric space.

  • ultrametrisable if it is homeomorphic to an ultrametric space.

  • zero-dimensional if every point has a basis of clopen neighbourhoods.

  • strongly zero-dimensional if for each closed subset $F$ and each neighbourhood $U$ of $F$, there is a clopen neighbourhood of $F$ contained in $U$.

Question. Is every zero-dimensional metrisable space ultrametrisable?

If I am not mistaken, this is true for compact spaces and more generally, for Lindelöf spaces. For the general case, according to an exercise in Bourbaki [General topology, Chapter 9], a metrisable space is ultrametrisable if and only if it is strongly zero-dimensional. Moreover, it is added in a footnote that it is not known whether every zero-dimensional metrisable space is strongly zero-dimensional.

Thus the problem was apparently open when Bourbaki published his volume on general topology, but I wonder whether it has been solved since then.

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Prabir Roy, Nonequality of Dimensions for Metric Spaces, Transactions of the American Mathematical Society Vol. 134, No. 1 (Oct., 1968), pp. 117-132, available here [PDF], has a complicated construction (which I have not gone through) of a complete metric space that is zero-dimensional but not strongly zero-dimensional.

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  • $\begingroup$ Thanks a lot! I accept your answer immediately, but indeed the construction in the paper looks particularly complicated. $\endgroup$ – J.-E. Pin Oct 17 at 23:53
  • $\begingroup$ @J.-E.Pin: You’re welcome! Yes, it looks downright formidable, though I suspect that the proof that the space is not strongly zero-dimensional may be worse. $\endgroup$ – Brian M. Scott Oct 17 at 23:55
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    $\begingroup$ Browsing on this site, I just discovered a possibly related answer to my question. Anyway, I will have a look at the given references. $\endgroup$ – J.-E. Pin Oct 18 at 17:42

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