A continuum with no prime shape factors |
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Authors: | Danuta Ko?odziejczyk |
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Institution: | Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warsaw, Poland |
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Abstract: | As the main result of this paper, we prove that there exists a continuum with non-trivial shape without any prime factor. This answers a question of K. Borsuk K. Borsuk, Concerning the notion of the shape of compacta, in: Proc. Internat. Symposium on Topology and Its Applications, Herceg-Novi, 1968, pp. 98-104]. We also show that for each integer n?3 there exists a continuum X such that Sh(X,x)=Shn(X,x), but Sh(X,x)≠Shn−1(X,x). Therefore we obtain the negative answer to another question of K. Borsuk K. Borsuk, Some remarks concerning the shape of pointed compacta, Fund. Math. 67 (1970) 221-240]: Does Sh(X,x)=Shn(X,x), for a compactum X and some integer n?3, implie that Sh(X,x)=Sh2(X,x)? |
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Keywords: | 55P55 |
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