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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dimension of the product and classical formulae of dimension theory


Authors: Alexander Dranishnikov and Michael Levin
Journal: Trans. Amer. Math. Soc. 366 (2014), 2683-2697
MSC (2010): Primary 55M10; Secondary 54F45, 55N45
Published electronically: September 26, 2013
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Abstract: Let $ f : X \longrightarrow Y$ be a map of compact metric spaces. A classical theorem of Hurewicz asserts that $ \dim X \leq \dim Y +\dim f$, where $ \dim f =\sup \{ \dim f^{-1}(y): y \in Y \}$. The first author conjectured that $ \dim Y + \dim f$ in Hurewicz's theorem can be replaced by $ \sup \{ \dim (Y \times f^{-1}(y)): y \in Y \}$. We disprove this conjecture. As a by-product of the machinery presented in the paper we answer in the negative the following problem posed by the first author: Can the Menger-Urysohn Formula $ \dim X \leq \dim A + \dim B +1$ for a decomposition of a compactum $ X=A\cup B$ into two sets be improved to the inequality $ \dim X \leq \dim (A \times B) +1$?

On a positive side we show that both conjectures hold true for compacta $ X$ satisfying the equality $ \dim (X\times X)=2\dim X$.


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Additional Information

Alexander Dranishnikov
Affiliation: Department of Mathematics, University of Florida, 444 Little Hall, Gainesville, Florida 32611-810
Email: dranish@math.ufl.edu

Michael Levin
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel
Email: mlevine@math.bgu.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05967-3
PII: S 0002-9947(2013)05967-3
Keywords: Dimension, cohomological dimension, Menger-Urysohn Formula, Hurewicz's Theorem
Received by editor(s): December 5, 2011
Received by editor(s) in revised form: September 9, 2012
Published electronically: September 26, 2013
Additional Notes: The first author was supported by NSF grant DMS-0904278; the second author was supported by ISF grant 836/08
Article copyright: © Copyright 2013 American Mathematical Society