Dimension in a reasonable class of spaces (Q1935850)

The first result of this very interesting paper is the following. Let \(f: X \to M\) be a light map from a compact space to a compact metric space. Then for every completely paracompact (in particular, Lindelöf) subspace \(Y\) of \(X\) the dimensions dim, ind, Ind, ind\(_0\), Ind\(_0\), \(\sigma\) and \(\Delta\) coincide. Then the author introduces a notion of a \textit{reasonable} class \(P\) of compact spaces which (a) is closed with respect to taking finite products, (b) is closed with respect to taking closed subsets, (c) contains all metrizable compacta, and (d) each member of \(P\) can be mapped onto a metrizable space by a light map. If one of members of such \(P\) contains the space \(J(\tau)^\mathbb N\), where \(J(\tau)\) is the hedgehog with \(\tau\) spines, then \(P\) is called \textit{very reasonable}. For each reasonable class \(P\) the following is proved. For any space \(X\) possessing a compactification \(aX\) in \(P\) there is a compactification \(bX\) in \(P\) such that \(bX \geq aX\) and dim bX = dim X. Furthermore, for a given map \(\pi : X \to Y\) from a locally compact space \(X\) to a compact space \(Y\) the author endows the disjoint union \(K(\pi)\) of \(X\) and \(Y\) with a compact topology such that the compact space \(K(\pi)\) contains \(X\) as an open set and \(Y\) as a closed one. Moreover, if \(X\) and \(Y\) are metrizable then \(K(\pi)\) is \(0\)-uniform Eberlein. Three interesting examples of such \(\pi\) and \(K(\pi)\) are presented. All of these \(K(\pi)\)'s are \(0\)-uniform Eberlein compactifications of the space \(J(\tau)\), and they can be mapped onto metrizable compacta by light maps. Two of the presented \(K(\pi)\)'s are first-countable. So the spaces \(K(\pi)^\mathbb N\) in each of the three examples are compactifications of the space \(J(\tau)^\mathbb N\), and the closed subspaces of \(K(\pi)^\mathbb N\) constitute very reasonable classes. All of this allows to strengthen several recent results due to Pasynkov and some other authors. For example, now each metrizable space \(X\) of weight \(\tau\) (resp. \(\leq c\) (continuum)) has a (resp. first countable and) \(0\)-uniform Eberlein compactification \(bX\) such that dim \(X\) = DIM \(bX\), where DIM is any of the dimensions dim, ind, Ind, ind\(_0\), Ind\(_0\), \(\sigma\), \(\Delta\), and wbX = \(\tau\). Even examples of \(0\)-uniform Eberlein compacta with dim \(<\) ind are suggested. Two questions concerning Eberlein compactifications are left open.