Mappings between ANRs that are fine homotopy equivalences

We extend Lacher's result [6,7] that a closed t/V°°-maρ between locally compact, finite dimensional ANRs is a fine homotopy equivalence to the case of arbitrary separable ANRs. It is hoped that this theorem will be useful in studying manifolds modelled on the Hubert Cube. (See [1], section PF3. Added in proof. See also [9]). A set A CX has property UV if for each open set U of X containing A, there is an open V, with A C V C U such that V is null-homotopic in U. A mapping /: X -» Y of X onto Y is a l/Vmap if for each y G F , f~) is a C/V subset of X. The mapping / is said to be closed if the image of every closed set is closed and proper if the inverse image of every compact set is compact. An absolute neighborhood retract for metric spaces is denoted an ANR. If a is a cover of Y and gι and g2 are maps of a space A into Y,g{ is α-near g2 if for each aEA there is a U Ea containing g\(a) and g2(a). The map gi is α-homotopic to gl9gι~g2, if there is a homotopy λ: A xI-*Y taking gι to g2 with the property that for each aEA there exists UEa containing λ({α}x/). A map /: X-» Y is a fine homotopy equivalence if for each open cover, α, of Y there exists a map g: Y-+X such