×

Certain sequences with compact closure. (English) Zbl 1169.54003

The authors deal with the question when a \(\beta\)-sequence in a \(T_2\)-space \(X\) has a compact closure. They show that every \(\beta\)-sequence has a feebly compact closure and, if \(X\) is monotonically normal, a compact closure. In regular spaces and in Tychonoff spaces, they give several equivalent conditions for the compactness of the closure of a \(\beta\)-sequence. Further, they express \(\beta\)-sequences with countable closure by fairly concrete forms and give several answers to the question when a \(\beta\)-space is a strong \(\beta\)-space [Y. Yajima, Houston J. Math. 33, No. 2, 531–540 (2007; Zbl 1243.54046)]. Examples show the sharpness of the results.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D30 Compactness
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E99 Topological spaces with richer structures
54G20 Counterexamples in general topology

Citations:

Zbl 1243.54046
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balogh, Z.; Rudin, M. E., Monotone normality, Topology Appl., 47, 115-127 (1992) · Zbl 0769.54022
[2] van Douwen, E. K., The integers and topology, (Kunen, K.; Vaughan, J. E., Handbook of Set-theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 111-167
[3] Engelking, R., General Topology (1989), Herdermann-Verlag: Herdermann-Verlag Berlin · Zbl 0684.54001
[4] Hodel, R. E., More spaces and -spaces, Pacific J. Math., 38, 641-652 (1971) · Zbl 0219.54024
[5] Ostaszewski, A., On countably compact, perfectly normal spaces, J. London Math. Soc., 14, 505-516 (1976) · Zbl 0348.54014
[6] Roitman, J., Basic \(S\) and \(L\), (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 295-326
[7] Stephenson, R. M., Symmetrizable spaces and separability, Topology Proc., 4, 589-599 (1979) · Zbl 0447.54028
[8] Weiss, W., Countably compact spaces and Martin’s Axiom, Canad. J. Math., 30, 243-249 (1978) · Zbl 0357.54019
[9] Yajima, Y., Strong \(β\)-spaces and their countable products, Houston J. Math., 33, 531-540 (2007) · Zbl 1243.54046
[10] Yajima, Y., Normal covers of infinite products and normality of \(Σ\)-products, Topology Appl., 154, 103-114 (2007) · Zbl 1109.54010
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.