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A space is said to be small weakly infinite dimensional if it has a basis -' such that the collection of finite unions of elements of _ 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 -+ 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.

Proper hereditary shape equivalences preserve small weak infinite-dimensionality