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Resumen:
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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 ?:X?Y is a shape morphism, then there exists exactly one function ?:?(X)??(Y) satisfying the following condition: If X0??(X) and Y0=?(X0) then there is a shape morphism ?0:X0?Y0 such that S[i(Y0,Y)]??0=??S[i(X0,X)], where S[i(Y0,Y)] is the shape morphism induced by the inclusion. Moreover, ? is continuous and for every compact set A??(X) there exists exactly one shape morphism ?:p?1(A)?q?1(?(A)) satisfying the following condition: S[i(q?1(?(A)),Y)]??=??S[i(p?1(A),X)]. (2) Let ?:X?Y be a shape morphism such that the induced map ?:?(X)??(Y) is a homeomorphism. If for each component X0 of X the unique shape morphism ?0:X0?Y0=?(X0) satisfying S[i(Y0,Y)]??0=??S[i(X0,X)] is an isomorphism, then ? is an isomorphism.
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