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 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 such that dim (f × g) = 1. We improve this result of Sternfeld showing that there exists a map 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 is a dense -subset of .
@article{bwmeta1.element.bwnjournal-article-fmv151i1p47bwm,
author = {Michael Levin},
title = {Bing maps and finite-dimensional maps},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {47-52},
zbl = {0860.54028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv151i1p47bwm}
}
Levin, Michael. Bing maps and finite-dimensional maps. Fundamenta Mathematicae, Tome 149 (1996) pp. 47-52. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv151i1p47bwm/
[00000] [1] R. H. Bing, Higher dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 653-663. | Zbl 0043.16804
[00001] [2] N. Brown, Continuous collections of higher dimensional indecomposable continua, Ph.D. thesis, University of Wisconsin, 1958.
[00002] [3] K. Kuratowski, Topology II, PWN, Warszawa, 1968.
[00003] [4] M. Levin and Y. Sternfeld, Atomic maps and the Chogoshvili-Pontrjagin claim, preprint.
[00004] [5] B. A. Pasynkov, On dimension and geometry of mappings, Dokl. Akad. Nauk SSSR 221 (1975), 543-546 (in Russian).
[00005] [6] Y. Sternfeld, On finite-dimensional maps and other maps with "small" fibers, Fund. Math. 147 (1995), 127-133. | Zbl 0833.54020
[00006] [7] H. Toruńczyk, Finite to one restrictions of continuous functions, Fund. Math. 75 (1985), 237-249. | Zbl 0593.54035