Compositions of simple maps

Jerzy Krzempek

Fundamenta Mathematicae (1999)

  • Volume: 162, Issue: 2, page 149-162
  • ISSN: 0016-2736

Abstract

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A map (= continuous function) is of order ≤ k if each of its point-inverses has at most k elements. Following [4], maps of order ≤ 2 are called simple.  Which maps are compositions of simple closed [open, clopen] maps? How many simple maps are really needed to represent a given map? It is proved herein that every closed map of order ≤ k defined on an n-dimensional metric space is a composition of (n+1)k-1 simple closed maps (with metric domains). This theorem fails to be true for non-metrizable spaces. An appropriate map on a Cantor cube of uncountable weight is described.

How to cite

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Krzempek, Jerzy. "Compositions of simple maps." Fundamenta Mathematicae 162.2 (1999): 149-162. <http://eudml.org/doc/212416>.

@article{Krzempek1999,
abstract = { A map (= continuous function) is of order ≤ k if each of its point-inverses has at most k elements. Following [4], maps of order ≤ 2 are called simple.  Which maps are compositions of simple closed [open, clopen] maps? How many simple maps are really needed to represent a given map? It is proved herein that every closed map of order ≤ k defined on an n-dimensional metric space is a composition of (n+1)k-1 simple closed maps (with metric domains). This theorem fails to be true for non-metrizable spaces. An appropriate map on a Cantor cube of uncountable weight is described. },
author = {Krzempek, Jerzy},
journal = {Fundamenta Mathematicae},
keywords = {composition; simple map; closed map; map of order ≤ k; finite-dimensional; zero-dimensional; Cantor cube; metric space; finite dimension},
language = {eng},
number = {2},
pages = {149-162},
title = {Compositions of simple maps},
url = {http://eudml.org/doc/212416},
volume = {162},
year = {1999},
}

TY - JOUR
AU - Krzempek, Jerzy
TI - Compositions of simple maps
JO - Fundamenta Mathematicae
PY - 1999
VL - 162
IS - 2
SP - 149
EP - 162
AB - A map (= continuous function) is of order ≤ k if each of its point-inverses has at most k elements. Following [4], maps of order ≤ 2 are called simple.  Which maps are compositions of simple closed [open, clopen] maps? How many simple maps are really needed to represent a given map? It is proved herein that every closed map of order ≤ k defined on an n-dimensional metric space is a composition of (n+1)k-1 simple closed maps (with metric domains). This theorem fails to be true for non-metrizable spaces. An appropriate map on a Cantor cube of uncountable weight is described.
LA - eng
KW - composition; simple map; closed map; map of order ≤ k; finite-dimensional; zero-dimensional; Cantor cube; metric space; finite dimension
UR - http://eudml.org/doc/212416
ER -

References

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  14. [14] K. Nagami, Mappings of finite order and dimension theory, Japan. J. Math. 30 (1960), 25-54. Zbl0106.16002
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