On the Smale theorem in weak shape theory

AbstractWe modify various lemmas from Dydak's paper on the Vietoris-Smale theorem to obtain more general results. We consider closed subsets X and Y of paracompact spaces M, N respectively, and a map F:M→N such that F∥X:X→Y is closed and surjective and X=F-1(Y) to obtain the following result.(Theorem). If F-1(y)ϵACnM(K) for each y ϵ Y and N is LCn+1, then the morphism in-pro-πk[F]:πkUKM(X,x)→πkUKN(Y,y) induced by the morphism [F]:UKM(X,x)→UKN(Y,y) is an isomorphism of in-pro-Grp for k⩽n and an epimorphism for k=n+1