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

Fundamenta Mathematicae (1995)

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

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topSternfeld, 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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- [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
- [K] K. Kuratowski, Topology II, PWN, Warszawa, 1968.
- [Le] M. Levin, A short construction of hereditarily infinite dimensional compacta, Topology Appl., to appear.
- [Pa] B. A. Pasynkov, On dimension and geometry of mappings, Dokl. Akad. Nauk SSSR 221 (1975), 543-546 (in Russian).
- [Po] R. Pol, On light mappings without perfect fibers on compacta, preprint. Zbl0913.54015
- [T] H. Toruńczyk, Finite to one restrictions of continuous functions, Fund. Math. 75 (1985), 237-249. Zbl0593.54035