# Bing maps and finite-dimensional maps

Fundamenta Mathematicae (1996)

• Volume: 151, Issue: 1, page 47-52
• ISSN: 0016-2736

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## Abstract

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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 $g:X\to {𝕀}^{k}$ 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 $g:X\to {𝕀}^{k}$ such that dim (f × g) = 1. We improve this result of Sternfeld showing that there exists a map $g:X\to mathbb{I}^{k+1}$ 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 $𝕀$ is a dense ${G}_{\delta }$-subset of $C\left(X,𝕀\right)$.

## How to cite

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Levin, Michael. "Bing maps and finite-dimensional maps." Fundamenta Mathematicae 151.1 (1996): 47-52. <http://eudml.org/doc/212182>.

@article{Levin1996,
abstract = {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 $C(X, \mathbb \{I\})$.},
author = {Levin, Michael},
journal = {Fundamenta Mathematicae},
keywords = {finite-dimensional maps; hereditarily indecomposable continua},
language = {eng},
number = {1},
pages = {47-52},
title = {Bing maps and finite-dimensional maps},
url = {http://eudml.org/doc/212182},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Levin, Michael
TI - Bing maps and finite-dimensional maps
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 1
SP - 47
EP - 52
AB - 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 $C(X, \mathbb {I})$.
LA - eng
KW - finite-dimensional maps; hereditarily indecomposable continua
UR - http://eudml.org/doc/212182
ER -

## References

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1. [1] R. H. Bing, Higher dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 653-663. Zbl0043.16804
2. [2] N. Brown, Continuous collections of higher dimensional indecomposable continua, Ph.D. thesis, University of Wisconsin, 1958.
3. [3] K. Kuratowski, Topology II, PWN, Warszawa, 1968.
4. [4] M. Levin and Y. Sternfeld, Atomic maps and the Chogoshvili-Pontrjagin claim, preprint.
5. [5] B. A. Pasynkov, On dimension and geometry of mappings, Dokl. Akad. Nauk SSSR 221 (1975), 543-546 (in Russian).
6. [6] Y. Sternfeld, On finite-dimensional maps and other maps with "small" fibers, Fund. Math. 147 (1995), 127-133. Zbl0833.54020
7. [7] H. Toruńczyk, Finite to one restrictions of continuous functions, Fund. Math. 75 (1985), 237-249. Zbl0593.54035

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