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On basic embeddings of compacta into the plane

Published online by Cambridge University Press:  17 April 2009

Neža Mramor-Kosta  and
Eva Trenklerová
Neža Mramor-Kosta
Affiliation:
Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Slovenia, e-mail: [email protected]
Eva Trenklerová
Affiliation:
Faculty of Science, P. J. Šafárik University Košice, Slovakia, e-mail: [email protected]
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Abstract

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A compactum K ⊂ ℝ2 is said to be basically embedded in ℝ2 if for each continuous function f: K → ℝ there exist continuous functions g, h: ℝ → ℝ such that f(x, y) = g(x) + h(y) for each point (x, y) ∈ K. Sternfeld gave a topological characterization of compacta K which are basically embedded in ℝ2 which can be formulated in terms of special sequences of points called arrays, using arguments from functional analysis. In this paper we give a simple topological proof of the implication: if there exists an array in K of length n for any n ∈ ℕ, then K is not basically embedded.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 3 , December 2003 , pp. 471 - 480
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Arnold, V.I., ‘On functions of three variables’, (in Russian), Dokl. Akad. Nauk. 114 (1957), 679–681.Google Scholar
[2]Arnold, V.I., ‘Problem 6’, Math. Ed. 3 (1958), 273–273.Google Scholar
[3]Arnold, V.I., ‘Representation of continuous functions of three variables by the superpositions of continuous functions of two variables’, (in Russian), Math. Sb. 48 (1959), 3–74.Google Scholar
[4]Flores, A., ‘Über m-dimensionale Komplexe, die im R2m+1 absolut selbsverschlungen sind’, Erg. Math. Kolloqu. 6 (1935), 4–6.Google Scholar
[5]Hurewicz, W. and Wallman, H., Dimension theory (Princetion University Press, Princeton, N.J., 1941).Google Scholar
[6]Kolmogorov, A.N., ‘On the representations of continuous functions of many variables by superpositions of continuous functions of one variable and addition’, (in Russian), Dokl. Akad. Nauk. 114 (1957), 953–956.Google Scholar
[7]Kolmogorov, A.N., ‘On the representations of continuous functions of many variables by superpositions of continuous functions fewer variables’, (in Russian), Dokl. Akad. Nauk 108 (1956), 179–182.Google Scholar
[8]Levin, M., ‘Dimension and superposition of continuous functions’, Israel J. Math. 70 (1990), 205–218.CrossRefGoogle Scholar
[9]Ostrand, P.A., ‘Dimension of metric spaces and Hilbert's problem 13’, Bull. Amer. Math. Soc. 71 (1965), 619–622.CrossRefGoogle Scholar
[10]Repovš, D. and Željko, M., ‘On basic embeddings into the plane’, (Preprint).Google Scholar
[11]Skopenkov, A., ‘A description of continua basically embeddable in R2’, Topology Appl. 65 (1995), 29–48.CrossRefGoogle Scholar
[12]Sternfeld, Y., ‘Dimension, superposition of functions and separation of points, in compact metric spaces’, Israel J. Math. 50 (1985), 13–52.CrossRefGoogle Scholar
[13]Sternfeld, Y., ‘Hilbert's 13th problem and dimension’, Lecture Notes in Math. 1376 (1989), 1–49.CrossRefGoogle Scholar
[14]Whitney, H., ‘The self-intersections of a smooth n-manifold in 2n-space’, Ann. of Math. (2) 45 (1944), 220–246.CrossRefGoogle Scholar
[15]van Kampen, E.R., ‘Komplexe in euklidischen Räumen’, Abh. Math. Sem. Univ. Hamburg 9 (1932), 72–78, 152–153.CrossRefGoogle Scholar