On the Smale theorem in weak shape theory
Weak shape, defined by \textit{K. Borsuk} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 18, 127-132 (1970; Zbl 0195.530)], coincides with compactly generated shape, defined by \textit{L. Rubin} and \textit{T. J. Sanders} [Gen. Topol. Appl. 4, 73-83 (1974; Zbl 0274.55012)], in locally compact metric spaces [\textit{T. J. Sanders}, Fundam. Math. 93, 37-40 (1976; Zbl 0339.55011)]. Compactly generated shape is defined using direct systems of compact spaces and shape morphisms. Therefore, its study reduces to the study of the category in-pro-C, where C is the homotopy category of polyhedra. Continuing the work of the second author [''The Hurewicz and Whitehead theorems with compact carriers'', Fundam. Math. (to appear)], the authors prove a Vietoris-Smale theorem for closed surjective maps \(F: X\to Y:\) If the fibers \(F^{-1}(y)\), \(y\in Y\), are approximately n-connected up to dimension n, then the induced morphism of the k-dimensional homotopy in-pro-groups is an isomorphism for \(k\leq n\) and an epimorphism for \(k=n+1\). This is more general than a similar result due to \textit{T. J. Sanders} [Fundam. Math. 118, 69-74 (1983; Zbl 0563.55008)] because F is not assumed to be proper. To obtain their result, the authors modify and generalize various results of \textit{J. Dydak} [Diss. Math. 156 (1979; Zbl 0405.55010)].
scientific article; zbMATH DE number 4007219