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Countable dimensionality and dimension raising cell-like maps. (English) Zbl 0913.54029

The setting is separable metric spaces. The term map is used to designate a continuous function. A map \(f:X\to Y\) is proper provided \(f^{-1}(C)\) is compact for every compact \(C\subset Y\). A proper map \(f\) is cell-like if each \(f^{-1}(y)\) has trivial shape. A proper map \(f\) is a hereditary shape equivalence if \(f| f^{-1}(A):f^{-1}(A)\to A\) is a shape equivalence for every closed subset \(A\subset Y\). Finally, a space \(X\) is countably dimensional provided it is the union of countable many finite-dimensional subspaces. It is easy to show that, for a hereditary shape equivalence \(f:X\to Y\), if \(X\) is finite dimensional, then \(Y\) is finite dimensional. It is unknown whether \(Y\) must be countably dimensional if \(X\) is countably dimensional. The paper presents some results that suggest that it may well be that countable dimensionality is not preserved by hereditary shape equivalences between complete spaces. Define \(\eta(X)=\sup\{\text{ind} Y:Y\) is a countably dimensional cell-like image of \(X\}\). In an earlier paper it was shown that if for every countably dimensional compactum \(X\), \(\eta(X)\) is countable, then hereditary shape equivalences between compacta preserve countable dimensionality. The paper contains the following version of a converse: if hereditary shape equivalences between separable, complete metric spaces preserve countable dimensionality, then for every countable ordinal \(\alpha\), \(\sup \{\eta (X):X\) is a compactum with \(\text{ind}(X)\leq\alpha\} <\omega_1\). Producing maps from an \(\omega\)-dimensional compactum to a compactum with \(\text{ind}> \alpha\) for any countable ordinal \(\alpha\) would show that countable dimensionality is not be preserved by hereditary shape equivalences.

MSC:

54F45 Dimension theory in general topology
55P10 Homotopy equivalences in algebraic topology
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