Extensional maps and approximate inverse limits
type: Item
instance of: scholarly article
title: Extensional maps and approximate inverse limits
zbMATH Open document ID: 1393.54010
author: Matthew Lynam
author: Leonard R. Rubin
published in: Topology and its Applications
publication date: 2018-03-27T00:00:00Z
review text: The notion of extensional equivalence (called extensional map in this paper) was introduced by \textit{Z. Virk} in [Topology Appl. 159, No. 3, 695--703 (2012; Zbl 1268.54012)] to generalize the factorization theorem of \textit{M. Levin, L. R. Rubin} and \textit{P. J. Schapiro} [Proc. Am. Math. Soc. 128, No. 10, 3099--3106 (2000; Zbl 0951.54033)]. A surjective map \(f: X\to Z\) is an extensional map relative to a space \(K\) if, given any pair \(A\subseteq B\) of closed subsets of \(Z\) and any map \(\alpha: A\to K\) for which \(\alpha\circ f|_{f^{-1}(A)}: f^{-1}(A) \to K\) extends over \(f^{-1}(B)\), the map \(\alpha\) extends over \(B\). In this paper, the authors consider this notion in the setting of inverse systems of compact Hausdorff spaces and approximate systems of compact metric spaces. In the first part of the paper, the authors prove: Let \({\boldsymbol Z} = (Z_\gamma, p_{\gamma\gamma^\prime}, \Gamma)\) be an inverse system of compact Hausdorff spaces, where \(\Gamma\) is a partially ordered directed set, and \(Z = \lim {\mathbf Z}\). For any surjective map \(f: X\to Z\) from a compact Hausdorff space \(X\) and any CW-complex \(K\), if \(p_\gamma\circ f: X\to Z\) is an extensional map relative \(K\) for each \(\gamma\in\Gamma\), then \(f: X\to Z\) is an extensional map relative \(K\). The main result of the paper is to prove an analogous result for an approximate system \({\boldsymbol Z} = \{Z_\gamma, \varepsilon_\gamma, p_{\gamma\gamma^\prime}, \Gamma\}\) of compact metric spaces where \(\Gamma\) is a partially ordered directed set that is directed, has no maximal element, and is cofinite. Using the Mardešić trick for approximate systems, the authors generalize the result for approximate systems with the index set being a partially ordered directed set that is directed and has no maximal element (not necessarily cofinite).
reviewed by: Takahisa Miyata
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 54C20
zbMATH DE Number: 6853369
zbMATH Keywords: absolute extensor
zbMATH Keywords: absolute neighborhood extensor
zbMATH Keywords: absolute neighborhood retract
zbMATH Keywords: approximate inverse system
zbMATH Keywords: cofinite
zbMATH Keywords: extensional map
zbMATH Keywords: extensional equivalence
zbMATH Keywords: inverse system
zbMATH Keywords: Mardešić trick
MaRDI profile type: Publication
full work available at URL: j.topol.2018.02.017
OpenAlex ID: W2792832220
cites work: Q5520521
cites work: Q5785434
cites work: Q4208405
cites work: The Mardesic factorization theorem for extension theory and c-separation
cites work: Approximate inverse systems of compacta and covering dimension
cites work: Cell-Like Mappings and Nonmetrizable Compacta of Finite Cohomological Dimension
cites work: Equivalence of the Borsuk and the ANR-system approach to shapes
cites work: Stability of gauged approximate resolutions
cites work: A generalization of the Levin-Rubin-Schapiro factorization theorem
DOI: 10.1016/J.TOPOL.2018.02.017
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description: scientific article; zbMATH DE number 6853369