Mappings of ANR's Whose Images are ANR's

Recently, R.J. Daverman and J.J. Walsh modified an example due to J. Taylor to obtain an example of a cell-like map from a compactum with non-trivial shape onto the Hilbert cube Q such that the non-degeneracy set is contained in the countable union of finite dimensional closed subsets of Q. Previously, G. Kozlowski proved that a cell-like map f: X' → X from a compact ANR X' onto a metric space X is a hereditary shape equivalence if there exists a sequence {B_n}∞_n=1 of finite dimensional closed subsets of X such that the non-degeneracy set is contained in ∞U_n=1 B_n and {f^-1 (B_n)}∞_n=1 forms a pairwise disjoint null-sequence. Here we raise two questions, which we show are equivalent. First: Is a cell-like map f: X' → X a hereditary shape equivalence if there exists a sequence {B_n}∞_n=1 of finite dimensional closed subsets of X such that U_n≠m (B_n ∩ B_m) has a strong transfinite dimension and the non-degeneracy set is contained in ∞U_n=1 B_n ? Even though we are not able to answer these questions, we give affirmative answers to the questions for special cases, and , furthermore, we are able to extend the aforementioned result of Kozlowski's. Also, we attempt to extend certain analyses of cell-like maps, which are proper by definition, to (non-proper) UV∞-maps. We prove that for a UV∞-map f: X' → X from an ANR X' to a metric space X the following are equivalent: (1) X is an ANR; (2) f is a hereditary homotopy equivalence; (3) f is a hereditary shape equivalence; (4) f is a fine homotopy equivalence. Since UV∞-maps are generally not onto, the notion of a "hereditary shape equivalence" is a variation of that for cell-like maps, though it agrees with, say, Kozlowski's for cell-like maps.