Miliaras, George; Sanderson, D. E. Complementary forms of \([\alpha,\beta]\)-compact. (English) Zbl 0827.54016 Topology Appl. 63, No. 1, 1-19 (1995). 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)]. Reviewer: J.E.Vaughan (Greensboro) Cited in 1 Document MSC: 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54D30 Compactness Keywords:finally \(\kappa\)-compact space; cofinally compact space; cofinally \(\kappa\)-compact space; \([\alpha, \beta]\)-compact space; cocompact space; cofinality of a space; normal almost compact spaces Citations:Zbl 0744.54010; JFM 55.0960.02 PDFBibTeX XMLCite \textit{G. Miliaras} and \textit{D. E. Sanderson}, Topology Appl. 63, No. 1, 1--19 (1995; Zbl 0827.54016) Full Text: DOI References: [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 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.