Cohomological dimension of Tychonov spaces
The concepts of dimension, including both covering (dim) and cohomological (dim\(_G\) for an abelian group \(G\)), have typically been studied for spaces which are at least normal, although the richest theory is usually developed for metrizable or at least paracompact spaces. A good source for dimension theory is [\textit{R. Engelking}, Theory of dimensions, finite and infinite (1995; Zbl 0872.54002)]; for cohomological dimension, one may consult [\textit{V. I. Kuzminov}, Russ. Math. Surv. 23, No. 5, 1-45 (1968); translation from Usp. Mat. Nauk 23, No. 5(143), 3-49 (1968; Zbl 0179.27903)]. The aim of the current paper is to extend the notion of cohomological dimension to the class of Tikhonov spaces which are not necessarily normal. The author points out that under the usual definition, dim\(_G X<\infty\) implies that \(X\) is normal. The definition chosen by this author agrees with the classical one for normal spaces, but does not force \(X\) to be normal. It is given in terms of extension of maps to \(K(G,n)\) as before, but in a certain different sense. The groups \(G\) considered must be countable, and \(K(G,n)\) is always taken as a Polish ANR-space homotopy equivalent to a corresponding Eilenberg-MacLane complex. One of the first results shows that dim\(_G X=\)dim\(_G\nu X\) where \(\nu X\) is the Hewitt realcompactification of \(X\). Some items proved are: (1) dim\(_\mathbb Z(A\cup B)\leq\) dim\(_\mathbb Z A+\) dim\(_\mathbb Z B+1\) (addition theorem, proved for metrizable spaces in [\textit{L. R. Rubin}, Topology Appl. 40, No. 3, 233-263 (1991; Zbl 0773.55002)]; (2) the existence of universal spaces of a given cohomological dimension and a given weight; (3) limit theorems; (4) a subspace theorem.
scientific article; zbMATH DE number 1067518
scientific article