Dimension of the product and classical formulae of dimension theory

Alexander Dranishnikov, Michael Levin

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Let f: X→y Y be a map of compact metric spaces. A classical theorem of Hurewicz asserts that dim X ≤ dim Y + dim f, where dim f = sup{dim f-1(y): y ε Y}. The first author conjectured that dim Y + dim f in Hurewicz's theorem can be replaced by sup{dim(Y × f-1(y)): y ε 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 ≤ dim A + dimB + 1 for a decomposition of a compactum X = A ∪ B into two sets be improved to the inequality dim X ≤ dim(A × B) + 1?. On a positive side we show that both conjectures hold true for compacta X satisfying the equality dim(X × X) = 2 dim X.

Original languageEnglish
Pages (from-to)2683-2697
Number of pages15
JournalTransactions of the American Mathematical Society
Volume366
Issue number5
DOIs
StatePublished - 25 Feb 2014

Keywords

  • Cohomological dimension
  • Dimension
  • Hurewicz's theorem
  • Menger-urysohn formula

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