Abstract
A closed set is called a cut between two disjoint sets if any continuum intersecting both these sets intersects the cut. The main result of this paper is that, for any compact space, the dimension defined by induction on the basis of the notion of cut does not exceed the covering dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
L. E. J. Brouwer, “Ñber den naturlichen Dimensionsbegriff,” J. Reine Angew. Math., 142 (1913), 146–152.
L. E. J. Brouwer, “Berichtung zur Abhandlung ‘Ñber den naturlichen Dimensionbegriff’ dieses Journal Bd. 142(1913), S. 146–152,” J. Reine Angew. Math., 153 (1924), 253.
W. Hurewicz and H. Wallman, Dimension Theory, London, 1941.
V. V. Fedorchuk and J. van Mill, “Dimensiongrad for locally connected Polish spaces,” Fund. Math., 163 (2000), no. 1, 77–82.
V. V. Fedorchuk, M. Levine, and E. V. Shchepin, “On Brouwer's definition of dimension,” Uspekhi Mat. Nauk [Russian Math. Surveys], 54 (1999), no. 2, 193–194.
P. S. Aleksandrov and B. A. Pasynkov, An Introduction to Dimension Theory [in Russian], Nauka, Moscow, 1973.
E. V. Shchepin, “Functors and uncountable powers of compact spaces,” Uspekhi Mat. Nauk [Russian Math. Surveys], 36 (1981), no. 3, 3–62.
S. Mardesic, “On covering dimension and inverse limits of compact spaces,” Illinois J. Math., 4 (1960), no. 2, 278–291.
V. V. Fedorchuk, “Compact spaces without intermediate dimensions,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 213 (1973), no. 4, 795–797.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fedorchuk, V.V. On the Brouwer Dimension of Compact Spaces. Mathematical Notes 73, 271–279 (2003). https://doi.org/10.1023/A:1022123528550
Issue Date:
DOI: https://doi.org/10.1023/A:1022123528550