On finite-dimensional maps and other maps with "small" fibers

Yaki Sternfeld

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 2, page 127-133
  • ISSN: 0016-2736

Abstract

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We prove that if f is a k-dimensional map on a compact metrizable space X then there exists a σ-compact (k-1)-dimensional subset A of X such that f|X∖A is 1-dimensional. Equivalently, there exists a map g of X in I k such that dim(f × g)=1. These are extensions of theorems by Toruńczyk and Pasynkov obtained under the additional assumption that f(X) is finite-dimensional.  These results are then extended to maps with fibers restricted to some classes of spaces other than the class of k-dimensional spaces. For example: if f has weakly infinite-dimensional fibers then dim(f|X∖A) ≤ 1 for some σ-compact weakly infinite-dimensional subset A of X. The proof applies essentially the properties of hereditarily indecomposable continua.

How to cite

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Sternfeld, Yaki. "On finite-dimensional maps and other maps with "small" fibers." Fundamenta Mathematicae 147.2 (1995): 127-133. <http://eudml.org/doc/212078>.

@article{Sternfeld1995,
abstract = {We prove that if f is a k-dimensional map on a compact metrizable space X then there exists a σ-compact (k-1)-dimensional subset A of X such that f|X∖A is 1-dimensional. Equivalently, there exists a map g of X in $I^k$ such that dim(f × g)=1. These are extensions of theorems by Toruńczyk and Pasynkov obtained under the additional assumption that f(X) is finite-dimensional.  These results are then extended to maps with fibers restricted to some classes of spaces other than the class of k-dimensional spaces. For example: if f has weakly infinite-dimensional fibers then dim(f|X∖A) ≤ 1 for some σ-compact weakly infinite-dimensional subset A of X. The proof applies essentially the properties of hereditarily indecomposable continua.},
author = {Sternfeld, Yaki},
journal = {Fundamenta Mathematicae},
keywords = {finite-dimensional map; weakly infinite-dimensional map; hereditarily indecomposable continuum; monotone map; Bing space},
language = {eng},
number = {2},
pages = {127-133},
title = {On finite-dimensional maps and other maps with "small" fibers},
url = {http://eudml.org/doc/212078},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Sternfeld, Yaki
TI - On finite-dimensional maps and other maps with "small" fibers
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 2
SP - 127
EP - 133
AB - We prove that if f is a k-dimensional map on a compact metrizable space X then there exists a σ-compact (k-1)-dimensional subset A of X such that f|X∖A is 1-dimensional. Equivalently, there exists a map g of X in $I^k$ such that dim(f × g)=1. These are extensions of theorems by Toruńczyk and Pasynkov obtained under the additional assumption that f(X) is finite-dimensional.  These results are then extended to maps with fibers restricted to some classes of spaces other than the class of k-dimensional spaces. For example: if f has weakly infinite-dimensional fibers then dim(f|X∖A) ≤ 1 for some σ-compact weakly infinite-dimensional subset A of X. The proof applies essentially the properties of hereditarily indecomposable continua.
LA - eng
KW - finite-dimensional map; weakly infinite-dimensional map; hereditarily indecomposable continuum; monotone map; Bing space
UR - http://eudml.org/doc/212078
ER -

References

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  1. [B] R. H. Bing, Higher dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 653-663. Zbl0043.16804
  2. [D-R] T. Dobrowolski and L. Rubin, The hyperspaces of infinite dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific J. Math. 104 (1994), 15-39. Zbl0801.54005
  3. [K] K. Kuratowski, Topology II, PWN, Warszawa, 1968. 
  4. [Le] M. Levin, A short construction of hereditarily infinite dimensional compacta, Topology Appl., to appear. 
  5. [Pa] B. A. Pasynkov, On dimension and geometry of mappings, Dokl. Akad. Nauk SSSR 221 (1975), 543-546 (in Russian). 
  6. [Po] R. Pol, On light mappings without perfect fibers on compacta, preprint. Zbl0913.54015
  7. [T] H. Toruńczyk, Finite to one restrictions of continuous functions, Fund. Math. 75 (1985), 237-249. Zbl0593.54035

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