Bing maps and finite-dimensional maps

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 -