×

Supercomplete topological spaces. (English) Zbl 1136.54016

Completeness in some form or another is always desirable: it guarantees that certain processes will converge. An important completeness property is Čech-completeness: the space is a \(G_\delta\)-set in its Čech-Stone compactification.
The authors define a stronger property, called supercompleteness: there is a sequence \(\langle\mathcal{U}_n\rangle_n\) of open covers such that every filter \(\mathcal{F}\), with the property that for every \(n\) the intersection \(\bigcap\{\operatorname{St}(F,\mathcal{U}_n:F\in\mathcal{F}\}\) belongs to \(\mathcal{F}\), has a cluster point.
Supercompleteness is equivalent to Čech-completeness in the class of paracompact spaces and also in the class of topological groups.
Supercompleteness is an inverse invariant of perfect maps and an invariant of open perfect maps. The ordinal space \(\omega_1\) is locally compact but not supercomplete. The authors introduce variants of supercompleteness by considering only filters of closed sets or filters with a countable base. All are equivalent to Čech-completeness in the class of metrizable spaces.
Reviewer: K. P. Hart (Delft)

MSC:

54D99 Fairly general properties of topological spaces
54F65 Topological characterizations of particular spaces
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D45 Local compactness, \(\sigma\)-compactness
54E50 Complete metric spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. V. Archangel’skii, On topological spaces which are complete in the sense of Čech, Vestn. Mosk. Univ. Ser. Mat., 2 (1961), 37–40 (in Russian).
[2] A. V. Archangel’skii, Bicompact sets and the topology of spaces, Trudy Moskov. Mat. Obšč., 13 (1965), 3–55. Trans. Moscow Math. Soc. (1965), 1–62.
[3] D. Buhagiar, Non locally compact points in ultracomplete topological spaces, Q & A in General Topology. 19 (2001), 125–131. · Zbl 0976.54025
[4] D. Buhagiar and I. Yoshioka, Ultracomplete topological spaces, Acta Math. Hungar., 92 (2001), 19–26. · Zbl 0997.54037 · doi:10.1023/A:1013743709044
[5] D. Buhagiar and I. Yoshioka, Sums and products in ultracomplete topological spaces, Top. Appl., 122 (2002), 77–86. · Zbl 1019.54015 · doi:10.1016/S0166-8641(01)00136-5
[6] B. S. Burdick, A note on completeness of hyperspaces, in: General Topology and Applications: Fifth Northeast Conference (Susan Andima et Al., ed.), Marcel Dekker (1991), pp. 19–24. · Zbl 0766.54008
[7] E. Čech, On bicompact spaces, Ann. of Math., 38 (1937), 823–844. · Zbl 0017.42803 · doi:10.2307/1968839
[8] Á. Császár, Strongly complete, supercomplete and ultracomplete spaces, in: Mathematical Structures – Computational Mathematics – Mathematical Modelling (Sofia), papers dedicated to Prof. L. Iliev’s 60th Anniversary (Sofia, 1975).
[9] J. Dugundji, Topology, Wm. C. Brown Publishers (Dubuque, Iowa, 1989).
[10] R. Engelking, General Topology, revised ed., Heldermann (Berlin, 1989).
[11] M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. del Circ. Mat. di Palermo, 22 (1906), 1–74. · JFM 37.0348.02 · doi:10.1007/BF03018603
[12] Z. Frolik, Generalization of the G {\(\delta\)} -property of complete metric spaces, Czech. Math. Journ., 10 (1960), 359–379.
[13] Z. Frolik, On the topological product of paracompact spaces, Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Math., Astro. et Phys., VIII (1960), 747–750.
[14] A. García-Máynez and S. Romaguera, Perfect pre-images of cofinally complete metric spaces, Comment. Math. Univ. Carolinae, 40 (1999), 335–342. · Zbl 0976.54032
[15] J. R. Isbell, Uniform Spaces (Providence, 1964).
[16] V. I. Ponomarev and V. V. Tkachuk, Countable character of X in {\(\beta\)}X versus the countable character of the diagonal in X {\(\times\)} X. Vestnik MGU, 5 (1987), 16–19 (in Russian). · Zbl 0637.54003
[17] B. A. Pasynkov, Almost metrizable groups, Dokl. Akad. Nauk SSSR, 161 (1965), 281–284 (in Russian). · Zbl 0132.27802
[18] B. A. Pasynkov, On open mappings, Dokl. Akad. Nauk SSSR, 175 (1967), 292–295 (in Russian), English translation: Soviet Math. Dokl., 8 (1967), 853–856. · Zbl 0155.50203
[19] S. Romaguera, On cofinally complete metric spaces, Q & A in General Topology, 16 (1998), 165–169. · Zbl 0941.54030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.