Proper hereditary shape equivalences preserve small weak infinite-dimensionality

Abstract:

A space is said to be small weakly infinite dimensional if it has a basis $B$ such that the collection of finite unions of elements of $B$ is inessential. A characterization of small weak infinite dimensionality is given for locally compact spaces. This characterization is then used to prove that if $f:X \to Y$ is a proper hereditary shape equivalence from a metric space $X$ which is small weakly infinite dimensional onto a locally compact metric space $Y$, then $Y$ is small weakly infinite dimensional.