On a theorem of B. J. Ball

Abstract

The author generalizes some results of Ball concerning the relationship between the shape of a locally compact metrizable space with compact components and the shape of its components. The following results are proved. Let X and Y be locally compact metrizable spaces with compact components. (1) If μ:X→Y is a shape morphism, then there exists exactly one function Λ:□(X)→□(Y) satisfying the following condition: If X0∈□(X) and Y0=Λ(X0) then there is a shape morphism μ0:X0→Y0 such that S[i(Y0,Y)]⋅μ0=μ⋅S[i(X0,X)], where S[i(Y0,Y)] is the shape morphism induced by the inclusion. Moreover, Λ is continuous and for every compact set A⊂□(X) there exists exactly one shape morphism η:p−1(A)→q−1(Λ(A)) satisfying the following condition: S[i(q−1(Λ(A)),Y)]⋅η=μ⋅S[i(p−1(A),X)]. (2) Let μ:X→Y be a shape morphism such that the induced map Λ:□(X)→□(Y) is a homeomorphism. If for each component X0 of X the unique shape morphism μ0:X0→Y0=Λ(X0) satisfying S[i(Y0,Y)]⋅μ0=μ⋅S[i(X0,X)] is an isomorphism, then μ is an isomorphism.