A theory of proper shape for locally compact metric spaces

On product of shape and a question of Sher

“…The concept was extended to arbitrary metrizable spaces by Borsuk [5] and by Fox [10], to locally compact metrizable spaces by Ball and Sher [1]. These extensions form different shape categories to each other.…”

“…ί^V) the strong (resp. weak) shape category in the sense of Borsuk [6], by ^F the shape category in the sense of Fox [10] and by <g* p the proper shape category in the sense of Ball and Sher [1], The shapes of a metrizable space X in <g^, <^w, <^F and ^v are denoted by Sh^X), Sh^X), Sh^X) and Sh^X) respectively.…”

“…Definition and notations* A familiarity with the basic terminology and notations of Borsuk's shape theory for metrizable spaces [6], of Fox's shape theory for metrizable spaces [10] and of Ball-Sher's proper shape theory for locally compact metrizable spaces [1] is assumed. A number of other technical or specialized definitions and notations are given in this section.…”

“…Ill] for metrizable spaces whose morphisms are the equivalence classes of strong (resp. weak) fundamental sequences, by ^F the shape category defined by Fox [10] for metrizable spaces whose morphisms are the equivalence classes of mutations and by ^ the proper shape category of Ball and Sher [1] for locally compact metrizable spaces whose morphisms are the equivalence classes of proper fundamental nets. (cf.…”

“…Consider the maps/, g: X->Ysuch that f(p) = (0, 0) and g(p) = (1,1). Then / and g induce non equivalent morphisms in the category <g^ or ^W f and induce equivalent morphisms in ^F or <^p.…”