Selections of multivalued maps and shape domination

Abstract

Given an approximate mapping f − ={f k }:X→Y between compacta from the Hilbert cube [K. Borsuk, Fund. Math. 62 (1968), 223–254, the author associates with f − a (u.s.c.) multivalued mapping F:X→Y . If F is single-valued, F and f − induce the same shape morphism, S(F)=S(f − ) . If Y is calm [Z. Čerin, Pacific J. Math. 79 (1978), no. 1, 69–91 and all F(x) , x∈X , are sufficiently small sets, then the existence of a selection for F implies that S(f − ) is generated by some mapping X→Y . If F is associated with f − and admits a coselection (a mapping g:Y→X such that y∈F(g(y)) , for y∈Y ), then S(f − ) is a shape domination and therefore sh(Y)≤sh(X) . If Y is even an FANR, then every sufficiently small multivalued mapping F:X→Y , which admits a coselection, induces a shape domination S(F) .