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No product of two nontrivial countable-dimensional continua maps lightly into any of the factors

Published online by Cambridge University Press:  07 August 2025

Roman Pol
Affiliation:
Institute of Mathematics, University of Warsaw, Warszawa 02-097, Poland e-mail: r.pol@mimuw.edu.pl
Mirosława Reńska*
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, Warszawa 00-662, Poland

Abstract

We shall prove that if X, Y are compact metrizable spaces of positive dimension and ${h:X\times Y \to X}$ is a continuous map with zero-dimensional fibers then X contains a nontrivial continuum without one-dimensional subsets; in particular, X is not a countable union of zero-dimensional sets, which provides a negative answer to a question of Dudák and Vejnar [DV]

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Article
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Aleksandrov, P. S., On some basic directions in general topology . Russian Math. Surv. 19(1964), 139.10.1070/RM1964v019n06ABEH001161CrossRefGoogle Scholar
Dudák, J. and Vejnar, B., Compact spaces homeomorphic to their respective squares . Eur. J. Math. 10(2024), no. 2, paper no. 37, 18 pp.CrossRefGoogle Scholar
Engelking, R., General topology. Helderman, Berlin, 1989.Google Scholar
Engelking, R., Theory of dimensions, finite and infinite. Sigma Series in Pure Mathematics, 10, Heldermann Verlag, Lemgo, 1995.Google Scholar
Grispolakis, J. and Tymchatyn, E. D., On confluent mappings and essential mappings—A survey . Rocky Mountain J. Math 11(1981), 131153.10.1216/RMJ-1981-11-1-131CrossRefGoogle Scholar
Henderson, D. W., A lower bound for transfinite dimension . Fund. Math. 63(1968), 167173.10.4064/fm-63-2-167-173CrossRefGoogle Scholar
Holsztyński, W., Universality of mappings onto products of snake-like spaces. Relation with dimension . Bull. Acad. Pol. Sci. 16(1968), 161167.Google Scholar
Hurewicz, W. and Wallman, H., Dimension theory. Princeton University Press, Princeton, 1948.Google Scholar
Levin, M., A dimensional property of Cartesian product . Fund. Math. 220(2013), 281286.CrossRefGoogle Scholar
van Mill, J., A Peano continuum homeomorphic to its own square, but not to its countable infinite product . Proc. Amer. Math. Soc. 80(1980), 703705.10.1090/S0002-9939-1980-0587960-0CrossRefGoogle Scholar
Pol, R., On classification of weakly infinite-dimensional compacta . Fund. Math. 116(1983), 169188.10.4064/fm-116-3-169-188CrossRefGoogle Scholar
Pol, R., Countable-dimensional universal sets . Trans. Amer. Math. Soc. 279(1986), 255268.Google Scholar
Reńska, M., On continua containing topologically their products with a nontrivial continuum, in preparation.Google Scholar
Walsh, J., Infinite-dimensional compacta containing no $n$ -dimensional ( $n\ge 1$ ) subsets . Topology 18(1979), 9195.CrossRefGoogle Scholar