TY - JOUR

T1 - Dimension of the product and classical formulae of dimension theory

AU - Dranishnikov, Alexander

AU - Levin, Michael

PY - 2014/2/25

Y1 - 2014/2/25

N2 - 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.

AB - 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.

KW - Cohomological dimension

KW - Dimension

KW - Hurewicz's theorem

KW - Menger-urysohn formula

UR - http://www.scopus.com/inward/record.url?scp=84894078361&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2013-05967-3

DO - 10.1090/S0002-9947-2013-05967-3

M3 - Article

AN - SCOPUS:84894078361

VL - 366

SP - 2683

EP - 2697

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -