Compositions of simple maps
Krzempek, Jerzy
Fundamenta Mathematicae, Tome 159 (1999), p. 149-162 / Harvested from The Polish Digital Mathematics Library
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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.

Publié le : 1999-01-01
EUDML-ID : urn:eudml:doc:212416 
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Krzempek, Jerzy. Compositions of simple maps. Fundamenta Mathematicae, Tome 159 (1999) pp. 149-162. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv162i2p149bwm/

[00000] [1] J. D. Baildon, Open simple maps and periodic homeomorphisms, Proc. Amer. Math. Soc. 39 (1973), 433-436. | Zbl 0268.54013

[00001] [2] C. Berge, Espaces topologiques. Fonctions multivoques, Dunod, Paris, 1959. | Zbl 0088.14703

[00002] [3] M. Bognár, On Peano mappings, Acta Math. Hungar. 74 (1997), 221-227. | Zbl 0924.54018

[00003] [4] K. Borsuk and R. Molski, On a class of continuous mappings, Fund. Math. 45 (1957), 84-98. | Zbl 0081.38803

[00004] [5] W. Dębski and J. Mioduszewski, Conditions which ensure that a simple map does not raise dimension, Colloq. Math. 63 (1992), 173-185. | Zbl 0757.54012

[00005] [6] J. Dydak, On elementary maps, ibid. 31 (1974), 67-69. | Zbl 0291.54008

[00006] [7] R. Engelking, Theory of Dimensions, Finite and Infinite, Heldermann, Lemgo, 1995. | Zbl 0872.54002

[00007] [8] R. Engelking, General Topology, PWN, Warszawa, 1977.

[00008] [9] R. Frankiewicz and W. Kulpa, On order topology of spaces having uniform linearly ordered bases, Comm. Math. Univ. Carolin. 20 (1979), 37-41. | Zbl 0411.54034

[00009] [10] W. Hurewicz, Über dimensionserhöhende stetige Abbildungen, J. Reine Angew. Math. 169 (1933), 71-78.

[00010] [11] M. Hušek and H. Ch. Reichel, Topological characterizations of linearly uniformizable spaces, Topology Appl. 15 (1983), 173-188.

[00011] [12] J. Krzempek, On decomposition of projections of finite order, Acta Univ. Carolin. Math. Phys. 36 (1995), 3-8. | Zbl 0826.54012

[00012] [13] A. Kucia and W. Kulpa, Spaces having uniformities with linearly ordered bases, Prace Nauk. Uniw. Śląskiego w Katowicach, Prace Mat. 3 (1973), 45-50. | Zbl 0255.54020

[00013] [14] K. Nagami, Mappings of finite order and dimension theory, Japan. J. Math. 30 (1960), 25-54. | Zbl 0106.16002

[00014] [15] J. H. Roberts, A theorem on dimension, Duke Math. J. 8 (1941), 565-574. | Zbl 0061.40101

[00015] [16] K. Sieklucki, On superposition of simple mappings, Fund. Math. 48 (1960), 217-228. | Zbl 0105.16405

[00016] [17] Yu. M. Smirnov, An example of a zero-dimensional space which has infinite covering dimension, Dokl. Akad. Nauk SSSR 123 (1958), 40-42 (in Russian).