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I have been looking through Alan Pears' "Dimension theory of general spaces" recently. In this book Pears references a 1960 paper by Aleksandrov and Ponomarev called "Some classes of $n$-dimensional spaces" that contains a sufficient condition for the large inductive dimension and the covering dimension to coincide for compact Hausdorff spaces. I can find the paper, but only in Russian.

Link to paper

Does anyone know what Aleksandrov and Ponomarev's condition is? Moreover, aside from being pseudometrizable, what are some conditions one can impose on a compact Hausdorff space $X$ so that $\operatorname{Ind}(X)=\dim(X)$?

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  • $\begingroup$ It looks like one such condition is that $dim(X)\leq n$ and there is either a metrizable space $Y$ and a continuous $f:X\rightarrow Y$ such that $dim(f)=0$ or else a perfectly zero dimensional space $Z$ and a continuous closed surjection $f:Z\rightarrow X$ such that the preimage of any $x\in X$ has at most $n+1$ points in $Z$. $\endgroup$
    – Robert Thingum
    Commented Oct 6, 2021 at 17:32
  • $\begingroup$ Engelking has a compact Hausdorff example by Lokucievskii that has $\dim(X)=1=\operatorname{ind}(X) < 2 = \operatorname{Ind}(X)$. $\endgroup$
    – Henno Brandsma
    Commented Oct 7, 2021 at 21:33

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You can read a review of the paper in Zentralblatt, it contains a short description in German.

The review on MathSciNet is a bit more extensive (but requires a subscription). There is indeed the condition of having an $n+1$-to-one map from a zero-dimensional compact space onto the space itself. There is also a condition on `gratings' (not defined, but my guess is, based on other papers: a grating is a finite closed cover where the interiors of the closed sets are pairwise disjoint).

In What is a non-metrizable analog of metrizable compacta? (Part I) Pasynkov defines a class of compacta where the dimensions coincide.

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  • $\begingroup$ Thank you for the reference, I will take a look! $\endgroup$
    – Robert Thingum
    Commented Oct 9, 2021 at 18:04

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