Extensional maps and approximate inverse limits
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).
scientific article; zbMATH DE number 6853369