Mapping cylinders of Hilbert cube factors

Abstract

It is shown that the class of Hilbert cube factors is closed under the operation of taking mapping cylinders and that the collapse-to-base of any such mapping cylinder generates in the “natural” way a uniform limit of homomorphisms between Hilbert cubes. From this is deduced as a corollary that the Cartesian product with X of any locally finite CW-complex is always an X-manifold if X is either the Hilbert cube or Hilbert space and that the countably infinite Cartesian product of non-degenerate, contractible, compact CW-complexes is always a Hilbert cube.