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](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=smj&paperid=4836&option_lang=eng)


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 $\operatorname{Ind}(X)=\dim(X)$?



Sufficient conditions for the covering dimension and large inductive dimension of compact Hausdorff spaces to coincide
You can read a review of the paper [in Zentralblatt](https://www.zbmath.org/?q=an%3A0108.35605), it contains a short description in German.


The review on [MathSciNet](https://mathscinet.ams.org/mathscinet-getitem?mr=124031) 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)](https://doi.org/10.1016/j.topol.2011.04.018) Pasynkov defines a class of compacta where the dimensions coincide.


