### Covariant and contravariant approaches to topology.

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CONTENTS§1. Introduction................................................................................................................................... 5§2. Some classes of objects and morphisms in pro-categories..................................................... 5§3. Shape category.................................................................................................................................... 14§4. Deformation dimension........................................................................................................................

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...

The purpose of this paper is to provide a geometric explanation of strong shape theory and to give a fairly simple way of introducing the strong shape category formally. Generally speaking, it is useful to introduce a shape theory as a localization at some class of “equivalences”. We follow this principle and we extend the standard shape category Sh(HoTop) to Sh(pro-HoTop) by localizing pro-HoTop at shape equivalences. Similarly, we extend the strong shape category of Edwards-Hastings to sSh(pro-Top)...

We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that $In{d}_{G}X=di{m}_{G}X$ if X is a separable metric ANR and G is a countable Abelian group. Hence $di{m}_{\mathbb{Z}}X=dimX$ for any separable metric ANR X.

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of $tow\left({H}_{0}\right)$ is an isomorphism if Y is movable. Recall that $\left({H}_{0}\right)$ is the full subcategory of $pro-{H}_{0}$ consisting of...

The Borsuk-Sieklucki theorem says that for every uncountable family ${{X}_{\alpha}}_{\alpha \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $dim\left({X}_{\alpha}\cap {X}_{\beta}\right)=n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $cl{c}_{\mathbb{Z}}^{n+1}$, where n ≥ 1, and G is an Abelian group. Let ${{X}_{\alpha}}_{\alpha \in J}$ be an uncountable family of closed subsets of X. If $di{m}_{G}X=di{m}_{G}{X}_{\alpha}=n$ for all α ∈ J, then $di{m}_{G}\left({X}_{\alpha}\cap {X}_{\beta}\right)=n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...

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