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A dimensional property of Cartesian product

Michael Levin (2013)

Fundamenta Mathematicae

We show that the Cartesian product of three hereditarily infinite-dimensional compact metric spaces is never hereditarily infinite-dimensional. It is quite surprising that the proof of this fact (and this is the only proof known to the author) essentially relies on algebraic topology.

Algebraic properties of quasi-finite complexes

M. Cencelj, J. Dydak, J. Smrekar, A. Vavpetič, Ž. Virk (2007)

Fundamenta Mathematicae

A countable CW complex K is quasi-finite (as defined by A. Karasev) if for every finite subcomplex M of K there is a finite subcomplex e(M) such that any map f: A → M, where A is closed in a separable metric space X satisfying XτK, has an extension g: X → e(M). Levin's results imply that none of the Eilenberg-MacLane spaces K(G,2) is quasi-finite if G ≠ 0. In this paper we discuss quasi-finiteness of all Eilenberg-MacLane spaces. More generally, we deal with CW complexes with finitely many...

Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin (2002)

Fundamenta Mathematicae

The Borsuk-Sieklucki theorem says that for every uncountable family X α α A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that d i m ( X α X β ) = n . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is c l c n + 1 , where n ≥ 1, and G is an Abelian group. Let X α α J be an uncountable family of closed subsets of X. If d i m G X = d i m G X α = n for all α ∈ J, then d i m G ( X α X β ) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...

Covering maps for locally path-connected spaces

N. Brodskiy, J. Dydak, B. Labuz, A. Mitra (2012)

Fundamenta Mathematicae

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow....

Dimension and ε -translations

Tatsuo Goto (1996)

Commentationes Mathematicae Universitatis Carolinae

Some theorems characterizing the metric and covering dimension of arbitrary subspaces in a Euclidean space will be obtained in terms of ε -translations; some of them were proved in our previous paper [G1] under the additional assumption of the boundedness of subspaces.

Estimates for homological dimension of configuration spaces of graphs

Jacek Świątkowski (2001)

Colloquium Mathematicae

We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b.

Extension theory of infinite symmetric products

Jerzy Dydak (2004)

Fundamenta Mathematicae

We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension ext-dim(X) was introduced by A. N. Dranishnikov [9] in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products SP(L). One of the main ideas of the paper is to treat ext-dim(X) ≤ SP(L) as the fundamental concept of cohomological dimension...

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