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Absorbing spaces for \(C\)-compacta. (English) Zbl 0930.54027

Recall that a space \(X\) is a \(C\)-space if each sequence \(\{\alpha_n\mid n\in\mathbb N\}\) of open covers admits an open cover which can be written as the countable union of pairwise disjoint collections \(\beta_n\) where each \(\beta_n\) refines \(\alpha_n\) [D. Addis and J. Gresham, Fundam. Math. 101, 195-205 (1978; Zbl 0397.54051)]. There is a notion \(\dim_C\), which is the transfinite extension of covering dimension that classifies \(C\)-compacta (spaces here are separable and metrizable). This concept was first defined by P. Borst and is recaptured in the current paper.
The author proves that for every countable ordinal \(\alpha\), there exists a \(C\)-compactum which is universal for the class of all compacta \(X\) with \(\dim_C X\leq\alpha\). The result is applied to show that for uncountably many ordinals \(\beta\), there exist noncountable-dimensional pre-Hilbert spaces \(D_\beta\) which are absorbing spaces [M. Bestvina and J. Mogilski, Mich. Math. J. 33, 291-313 (1986; Zbl 0629.54011)] for the class of compacta with \(\dim_C<\beta\).
Reviewer: L.R.Rubin (Norman)

MSC:

54F45 Dimension theory in general topology
57N17 Topology of topological vector spaces
57N20 Topology of infinite-dimensional manifolds
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References:

[1] P. Borst, Some remarks concerning \(C\); P. Borst, Some remarks concerning \(C\) · Zbl 1116.54018
[2] Bestvina, M.; Mogilski, J., Characterizing certain incomplete infinite-dimensional absolute retracts, Michigan Math. J., 33, 291-313 (1986) · Zbl 0629.54011
[3] Dobrowolski, T.; Mogilski, J., Problems on topological classification of incomplete metric spaces, (van Mill, J.; Reed, G. M., Open Problems in Topology (1990), North-Holland: North-Holland Amsterdam) · Zbl 0511.57009
[4] Dobrowolski, T.; Mogilski, J., Absorbing sets in the Hilbert cube related to transfinite dimension, Bull. Polish Acad. Sci., 38, 185-188 (1990) · Zbl 0782.57012
[5] M. Zarichnyi, Soft maps and their applications in infinite-dimensional topology, Preprint.; M. Zarichnyi, Soft maps and their applications in infinite-dimensional topology, Preprint.
[6] Addis, D. F.; Gresham, J. H., A class of infinite dimensional spaces. Part 1: Dimension theory and Alexandroff’s problem, Fund. Math., 101, 195-205 (1978) · Zbl 0397.54051
[7] Hattori, Y.; Yamada, K., Closed pre-images of \(C\)-spaces, Math. Japon., 34, 555-561 (1989) · Zbl 0694.54028
[8] Chatyrko, V., Classification of compacta with property \(C\), Questions Answers Gen. Topology, 10, 154-159 (1992) · Zbl 0794.54035
[9] Pol, R., On classification of weakly infinite-dimensional compacta, Fund. Math., 116, 169-188 (1983) · Zbl 0571.54030
[10] Lelek, A., On dimension of remainders in compactifications, Dokl. Akad. Nauk SSSR, 160, 534-537 (1965)
[11] Bessaga, C.; Pełczyński, A., Selected Topics in Infinite-dimensional Topology (1975), PWN: PWN Warsaw · Zbl 0304.57001
[12] Dobrowolski, T., Extending homeomorphisms and application to metric linear spaces without completeness, Trans. Amer. Math. Soc., 313, 753-784 (1989) · Zbl 0692.57007
[13] Pol, R., A weakly infinite-dimensional compactum which is not countable-dimensional, (Proc. Amer. Math. Soc., 82 (1981)), 634-636 · Zbl 0469.54014
[14] Engelking, R.; Pol, E., Countable-dimensional spaces: a survey, Diss. Math., 216, 1-45 (1983) · Zbl 0496.54032
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