Approximate inverse systems which admit meshes
The first author and \textit{L. R. Rubin} [Pac. J. Math. 138, No. 1, 129-144 (1989; Zbl 0661.54016)] introduced what are now called gauged approximate inverse systems that are subject to three axioms (A1), (A2) and (A3). The reviewer [Commentat. Math. Univ. Carol. 32, No. 3, 551-565 (1991; Zbl 0785.54016)], initiated the study of approximate inverse systems \((X_\alpha, p_{\alpha \beta}, A)\) that are required to satisfy only (A2): the difference between \(p_{\alpha \gamma}\) and \(p_{\alpha \beta} p_{\beta \gamma}\) can be made arbitrarily small provided \(\beta\) and \(\gamma\) are chosen to be sufficiently large. In gauged approximate systems the spaces \(X_\alpha\) are equipped with normal covers (meshes) \(U_\alpha\) that satisfy (A1) and (A3). While (A1) is always achieved by a slight modification of the order of the directed set \(A\), for each normal cover \(U\) of \(X_\alpha\), (A3) requires \(U_\beta\) to refine \(p_{\alpha \beta}^{- 1} (U)\) for all sufficiently large \(\beta\). Given an approximate system, the authors obtain a sufficient condition for the existence of meshes that satisfy (A3). In certain cases, the condition is also necessary.
scientific article; zbMATH DE number 681619
scientific article