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## Fundamenta Mathematicae

1999 | 162 | 2 | 149-162
Tytuł artykułu

### Compositions of simple maps

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
149-162
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-01-08
poprawiono
1998-10-05
poprawiono
1999-06-24
Twórcy
autor
• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland
Bibliografia
• [1] J. D. Baildon, Open simple maps and periodic homeomorphisms, Proc. Amer. Math. Soc. 39 (1973), 433-436.
• [2] C. Berge, Espaces topologiques. Fonctions multivoques, Dunod, Paris, 1959.
• [3] M. Bognár, On Peano mappings, Acta Math. Hungar. 74 (1997), 221-227.
• [4] K. Borsuk and R. Molski, On a class of continuous mappings, Fund. Math. 45 (1957), 84-98.
• [5] W. Dębski and J. Mioduszewski, Conditions which ensure that a simple map does not raise dimension, Colloq. Math. 63 (1992), 173-185.
• [6] J. Dydak, On elementary maps, ibid. 31 (1974), 67-69.
• [7] R. Engelking, Theory of Dimensions, Finite and Infinite, Heldermann, Lemgo, 1995.
• [8] R. Engelking, General Topology, PWN, Warszawa, 1977.
• [9] R. Frankiewicz and W. Kulpa, On order topology of spaces having uniform linearly ordered bases, Comm. Math. Univ. Carolin. 20 (1979), 37-41.
• [10] W. Hurewicz, Über dimensionserhöhende stetige Abbildungen, J. Reine Angew. Math. 169 (1933), 71-78.
• [11] M. Hušek and H. Ch. Reichel, Topological characterizations of linearly uniformizable spaces, Topology Appl. 15 (1983), 173-188.
• [12] J. Krzempek, On decomposition of projections of finite order, Acta Univ. Carolin. Math. Phys. 36 (1995), 3-8.
• [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.
• [14] K. Nagami, Mappings of finite order and dimension theory, Japan. J. Math. 30 (1960), 25-54.
• [15] J. H. Roberts, A theorem on dimension, Duke Math. J. 8 (1941), 565-574.
• [16] K. Sieklucki, On superposition of simple mappings, Fund. Math. 48 (1960), 217-228.
• [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).
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