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1996 | 151 | 1 | 47-52
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Bing maps and finite-dimensional maps

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Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map $g : X → \mathbb{I}^k$ such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open.
 Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map $g : X → \mathbb{I}^k$ such that dim (f × g) = 1. We improve this result of Sternfeld showing that there exists a map $g : X → mathbb{I}^{k+1}$ such that dim (f × g) =0. The last result is generalized to maps f with weakly infinite-dimensional fibers.
 Our proofs are based on so-called Bing maps. A compactum is said to be a Bing compactum if its compact connected subsets are all hereditarily indecomposable, and a map is said to be a Bing map if all its fibers are Bing compacta. Bing maps on finite-dimensional compacta were constructed by Brown [2]. We construct Bing maps for arbitrary compacta. Namely, we prove that for a compactum X the set of all Bing maps from X to $\mathbb{I}$ is a dense $G_δ$-subset of $C(X, \mathbb{I})$.
Rocznik
Tom
151
Numer
1
Strony
47-52
Opis fizyczny
Daty
wydano
1996
otrzymano
1996-01-04
poprawiono
1996-05-19
Twórcy
Bibliografia
  • [1] R. H. Bing, Higher dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 653-663.
  • [2] N. Brown, Continuous collections of higher dimensional indecomposable continua, Ph.D. thesis, University of Wisconsin, 1958.
  • [3] K. Kuratowski, Topology II, PWN, Warszawa, 1968.
  • [4] M. Levin and Y. Sternfeld, Atomic maps and the Chogoshvili-Pontrjagin claim, preprint.
  • [5] B. A. Pasynkov, On dimension and geometry of mappings, Dokl. Akad. Nauk SSSR 221 (1975), 543-546 (in Russian).
  • [6] Y. Sternfeld, On finite-dimensional maps and other maps with "small" fibers, Fund. Math. 147 (1995), 127-133.
  • [7] H. Toruńczyk, Finite to one restrictions of continuous functions, Fund. Math. 75 (1985), 237-249.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv151i1p47bwm
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