Approximative shape. I: Basic notions
type: Item
instance of: scholarly article
title: Approximative shape. I: Basic notions
zbMATH Open document ID: 0646.55006
author: Tadashi Watanabe
published in: Tsukuba Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Resolutions of spaces and maps [introduced by \textit{S. Mardešić} in Fund. Math. 114, 53-78 (1981; Zbl 0411.54019)] are important tools in shape theory. A problem, however, is that a map \(f: X\to Y\) in general fails to have a resolution \(f: {\mathcal X}\to {\mathcal Y}\) with respect to arbitrary polyhedral resolutions \(p: X\to {\mathcal X}\) and \(q: Y\to {\mathcal Y}\). The present paper solves this problem by introducing the concept of approximative resolutions. Let C be a subcategory of Top, e.g. the full subcategory AP of approximate polyhedra. The approximative pro-category Appro-C is defined as follows. The objects are approximative inverse systems (\({\mathcal X},{\mathcal U})\) over C consisting of a cofinite inverse system \({\mathcal X}=\{X_{\alpha}\}\) over C plus a system \({\mathcal U}=\{U_{\alpha}\}\) of normal open coverings \(U_{\alpha}\) of \(X_{\alpha}\) such that certain refinement conditions are fulfilled. The morphisms are suitable equivalence classes of approximative system maps (the latter differ from maps of inverse systems in replacing the standard commutativity conditions by closeness conditions with respect to the coverings). An approximative C-resolution of a space X is an object (\({\mathcal X},{\mathcal U})\) of Appro-C together with an ordinary resolution \(p: X\to {\mathcal X}\) (for \(C=Top\), one speaks about approximative resolutions). An approximative resolution of a map \(f: X\to Y\) with respect to approximative resolutions \(p: X\to ({\mathcal X},{\mathcal U})\) and \(q: Y\to ({\mathcal Y},{\mathcal V})\) is an approximative system map \(f: ({\mathcal X},{\mathcal U})\to ({\mathcal Y},{\mathcal V})\) such that \(qf\) and \(fp\) are \({\mathcal V}\)-close. One of the main results is that for any map \(f: X\to Y\), any approximative resolution \(p: X\to ({\mathcal X},{\mathcal U})\) and any approximative AP-resolution \(q: Y\to ({\mathcal Y},{\mathcal V})\), there exists an approximative resolution \(f: ({\mathcal X},{\math…
reviewed by: Peter Mrozik
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 54C56
zbMATH DE Number: 4054600
zbMATH Keywords: Tikhonov functor
zbMATH Keywords: Resolutions of spaces and maps
zbMATH Keywords: shape theory
zbMATH Keywords: approximative resolutions
zbMATH Keywords: approximate polyhedra
zbMATH Keywords: approximative pro- category
zbMATH Keywords: approximative inverse systems
zbMATH Keywords: approximative resolution of a map
zbMATH Keywords: approximative shape category
zbMATH Keywords: Tychonoff functor
zbMATH Keywords: completion functor
zbMATH Keywords: realization functor
zbMATH Keywords: topologically complete Tychonoff spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160501
OpenAlex ID: W268961749
DOI: 10.21099/TKBJM/1496160501
description: scientific article; zbMATH DE number 4054600