The quotient shapes of vectorial spaces are considered-algebraically and topologically, especially, of the normed spaces. In the algebraic case, all the shape classifications and the isomorphism classification coincide. However, in the general topological case and, especially, in the normed case, the quotient shape classifications are strictly coarser than the isomorphism classification.

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Abstract.We show under what conditions, and how, one can obtain a shape theory (various shape theories) in a concrete category. The technique is, roughly speaking, reduced to the quotients by congruences providing the objects of lower cardinalities. The application yields the new (coarser) classifications in every concrete category which admits sufficiently many non-trivial quotients. Thus, the ordering, (ultra)pseudometric, uniform and topological structures, as well as many algebraic and mixed (multi-) structures, give rise to interesting results.

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Abstract. By reducing the Mardešić S-equivalence to a finite case, i.e., to each n ∈ {0} ∪ N separately, we have derived the notions of Snequivalence and S n+1 -domination of compacta. The Sn-equivalence for all n coincides with the S-equivalence. Further, the S n+1 -equivalence implies S n+1 -domination, and the S n+1 -domination implies Sn-equivalence. The S 0 -equivalence is a trivial equivalence relation, i.e., all non empty compacta are mutually S 0 -equivalent. It is proved that the S 1 -equivalence is strictly finer than the S 0 -equivalence, and that the S 2 -equivalence is strictly finer than the S 1 -equivalence. Thus, the S-equivalence is strictly finer than the S 1 -equivalence. Further, the S 1 -equivalence classifies compacta which are homotopy (shape) equivalent to ANR's up to the homotopy (shape) types. The S 2 -equivalence class of an FANR coincides with its S-equivalence class as well as with its shape type class. Finally, it is conjectured that, for every n, there exists n > n such that the S n -equivalence is strictly finer than the Sn-equivalence.