engleski

The S_n-equivalence of compacta

By reducing the Mardešić S-equivalence to a finite case, i.e. to each n∈ {;0};∪ N separately, we have derived the notions of S_{;n};-equivalence and S_{;n+1};-domination of compacta. The S_{;n};-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 S_{;n};-equivalence. The S₀ -equivalence is a trivial equivalence relation, i.e. all non empty compacta are mutually S₀ -equivalent. It is proved that the S₁ -equivalence is strictly finer than the S₀ -equivalence, and that the S₂ -equivalence is strictly finer than the S₁ -equivalence. Thus, the S₁ -equivalence is strictly coarser than the S-equivalence. Further, the S₁ -equivalence classifies compacta which are homotopy (shape) equivalent to ANR's up to the homotopy types (shape types). The S₂ -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 an n′ >n such that the S_{;n′ };-equivalence is strictly finer than the S_{;n};-equivalence.

Compactum; ANR; shape; S-equivalece

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