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Approximate inverse systems which admit meshes. (English) Zbl 0828.54010

The first author and 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.

MSC:

54B35 Spectra in general topology
54F45 Dimension theory in general topology
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References:

[1] Charalambous, M. G., Approximate inverse systems of uniform spaces and an application of inverse systems, Comment. Math. Univ. Carolin., 32, 551-565 (1991) · Zbl 0785.54016
[2] Engelking, R., General Topology, (Monografie Matematyczne, 60 (1977), PWN: PWN Warsaw) · Zbl 0684.54001
[3] Mardešić, S., Inverse limits and resolutions, (Shape Theory and Geometric Topology. Shape Theory and Geometric Topology, Lecture Notes in Mathematics, 870 (1981), Springer: Springer Berlin), 239-252, Proceedings (Dubrovnik 1981)
[4] Mardešić, S., Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math., 114, 53-78 (1981) · Zbl 0411.54019
[5] Mardešić, S., On approximate inverse systems and resolutions, Fund. Math., 142, 241-255 (1993) · Zbl 0810.54013
[6] Mardešić, S.; Rubin, L. R., Approximate inverse systems of compacta and covering dimension, Pacific J. Math., 138, 129-144 (1989) · Zbl 0631.54006
[7] Mardešić, S.; Rubin, L. R., Cell-like mappings and nonmetrizable compacta of finite cohomological dimension, Trans. Amer. Math. Soc., 313, 53-79 (1989) · Zbl 0698.54027
[8] Mardešić, S.; Rubin, L. R.; Uglešić, N., A note on approximate systems of metric compacta, Topology Appl., 59, 189-194 (1994) · Zbl 0855.54015
[9] Mardešić, S.; Segal, J., Shape Theory (1982), North-Holland: North-Holland Amsterdam · Zbl 0495.55001
[10] Mardešić, S.; Watanabe, T., Approximate resolutions of spaces and mappings, Glas. Mat., 24, 583-633 (1989) · Zbl 0715.54009
[12] Morita, K., Resolutions of spaces and proper inverse systems in shape theory, Fund. Math., 124, 263-270 (1984) · Zbl 0564.55006
[14] Watanabe, T., Approximative shape I, Tsukuba J. Math., 11, 17-59 (1987) · Zbl 0646.55006
[15] Watanabe, T., Approximate resolutions and covering dimension, Topology Appl., 38, 147-154 (1991) · Zbl 0716.54021
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