Abstract
A concept of relative or reduced mapping cylinders is introduced, and it is shown that if ƒ is any map between two spaces which have the property that their products with the Hilbert cube are compact Hilbert cube manifolds, then for any closed subset A of the domain of ƒ the relative mapping cylinder of ƒ reduced modulo A also has this property and that the two collapses associated with it generate in a natural way near-homeomorphisms of Hilbert cube manifolds.
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