Shape properties of nonmetrizable spaces (Q1313912)

It is proved that a Hausdorff compactum can be represented as the inverse limit of a so-called homotopically stable \(\omega\)-inverse system (\(\omega\)-spectrum) \(S_ X = \{X_ a, p^ b_ a, A\}\). The term \(\omega\)-inverse system includes the following: (a) all \(X_ a\) are metric compacta; (b) \(S_ X\) is \(\omega\)-continuous (see the paper for the definition); (c) A has property that each countable chain \(B\) in \(A\) has \(\sup B\) in \(A\), while homotopically stable means that the projections \(p_ a : X \to X_ a\) have the following property: for each closed subset \(F\) of \(X_ a\), for each metrizable ANR-compactum \(Y\) and for any two maps, \(f, g:F \to Y\), the relation \(fp_ a | p^{-1}_ a (F) \simeq gp_ a | p^{-1}_ a (F)\) implies \(f \simeq g\). This result and related investigations are then applied first to the problem of metric shape, so that Theorem 5 of [\textit{T. Watanabe}, Fundam. Math. 104, 1-11 (1979; Zbl 0433.54027)] is reformulated. Then Theorem 1 of [\textit{R. Geoghegan} and \textit{R. C. Lacher}, ibid. 92, 25-27 (1976; Zbl 0339.55012)] is extended to Hausdorff compacta. It is also proved that for Hausdorff compacta the following conditions are equivalent: (i) for each compact \(G_ \delta\)-subset \(Z\) of \(X\) and each metrizable ANR-compactum \(P\) the set \([Z, P]\) is countable; (ii) \(X\) can be represented as the inverse limit of some \(\omega\)-inverse system all of whose projections are hereditary shape equivalences.