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Complementary forms of \([\alpha,\beta]\)-compact. (English) Zbl 0827.54016

Final \(\kappa\)-compactness is one of the classical open covering properties introduced by P. Alexandroff and P. Urysohn [Verh. Koninkl. Akad. Wetensch. Amsterdam 14, 1-96 (1929; JFM 55.0960.02)], and is a special case of their more general \([\alpha, \beta]\)- compactness. A space is called finally \(\kappa\)-compact provided every open cover has a subcover of cardinality less than \(\kappa\). Three main definitions in this paper are (1) a space \(X\), which is not finally \(\kappa\)-compact, is called cofinally \(\kappa\)-compact provided the set of all closed, not finally \(\kappa\)-compact subsets of \(X\), is linked (i.e., given two disjoint closed subsets of \(X\), one is finally \(\kappa\)- compact), (2) the cardinal invariant \(cfc(X)\) is defined to be the least cardinal \(\kappa\) such that \(X\) is cofinally \(\kappa\)-compact, and (3) the cofinality of a space \(X\) is the least cardinality of a closed filterbase \(\mathcal F\) in \(X\) with \(\bigcap {\mathcal F} = \emptyset\). The authors present basic results about these and related properties, and show how these notions are related to several properties in the literature. For example, cofinally \(\omega\)-compact \(T_3\)-spaces were studied by W. Fleissner, J. Kulesza and R. Levy under the name normal almost compact spaces [Proc. Am. Math. Soc. 113, No. 2, 503-511 (1991; Zbl 0744.54010)].

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness
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[1] Alexandroff, P. S.; Urysohn, P., Memoire sur les espaces topologiques compacts, Verh. Koninkl. Akad. Wetensch. Amsterdam, 14, 1-96 (1929) · JFM 55.0960.02
[2] Bell, M. G., Polyadic spaces of arbitrary compactness numbers, Comment. Math. Univ. Carolin., 26, 353-361 (1985) · Zbl 0587.54039
[3] Fleissner, W. G.; Kulesza, J.; Levy, R., Cofinality in normal almost compact spaces, (Proc. Amer. Math. Soc., 113 (1991)), 503-511 · Zbl 0744.54010
[4] Hodel, R. E.; Vaughan, J. E., A note on [a, b]-compactness, Gen. Topology Appl., 4, 179-189 (1974) · Zbl 0284.54012
[5] Miliaras, G., Cardinal invariants and covering properties in topology, (Thesis (1988), Iowa State University: Iowa State University Ames, IA)
[6] Miliaras, G., Initially compact and related spaces, Period. Math. Hungar., 24, 135-141 (1992) · Zbl 0780.54024
[7] Nyikos, P., The theory of nonmetrizable manifolds, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 603-683
[8] Rudin, M. E., Dowker spaces, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 761-780 · Zbl 0566.54009
[9] Stephenson, R. M., Initially κ-compact and related spaces, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 603-632 · Zbl 0588.54025
[10] Willard, S., General Topology (1968), Addison-Wesley: Addison-Wesley Cambridge, MA · Zbl 0205.26601
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