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Star covering properties. (English) Zbl 0743.54007
The authors provide a hierarchy of star-covering properties which fit between countable compactness and pseudocompactness: \(n\)-starcompact, strongly \(n\)-starcompact, and a newly defined property, \(\omega\)- starcompact. Lindelöf analogues are also discussed (\(n\)-star- Lindelöf, strongly \(n\)-star-Lindelöf, and a new \(\omega\)-star- Lindelöf), which rank between Lindelöf and the discrete countable chain condition.
The implications of these star-covering properties are also examined in the presence of regularity, and further, for Moore spaces. The authors give a helpful collection of counterexamples to differentiate among the various properties.

MSC:
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A35 Consistency and independence results in general topology
54E30 Moore spaces
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[1] Bagley, R.W.; Connell, E.H.; McKnight, J.D., On properties characterizing pseudocompact spaces, Proc. amer. math. soc., 9, 500-506, (1958) · Zbl 0089.17601
[2] Blair, R.L., Chain conditions in para-Lindelöf and related spaces, Topology proc., 11, 247-266, (1986) · Zbl 0642.54004
[3] Burke, D.K., Covering properties, (), 347-422
[4] van Douwen, E.K., The integers and topology, (), 111-167
[5] van Douwen, E.K.; Reed, G.M., On chain conditions in Moore spaces II, Topology appl., 39, 65-69, (1991), this issue. · Zbl 0727.54013
[6] Engelking, R., General topology, (1977), PWN Warsaw
[7] Fleischman, W.M., A new extension of countable compactness, Fund. math., 67, 1-9, (1970) · Zbl 0194.54601
[8] Fleissner, W.G., Normal Moore spaces in the constructible universe, Proc. amer. math. soc., 46, 294-298, (1974) · Zbl 0314.54028
[9] Gillman, L.; Jerison, M., Rings of continuous functions, () · Zbl 0151.30003
[10] R.W. Heath and G.M. Reed, Discrete countable chain condition T3-spaces with a σ-locally countable base which are not Lindelöf, to appear.
[11] Ikenaga, S., A class which contains Lindelöf spaces, separable spaces and countably compact spaces, Mem. numazu college tech., 18, 105-108, (1983)
[12] Ikenaga, S., Some properties of ω-n-star spaces, Res. rep. Nara nat. college tech., 23, 53-57, (1987)
[13] Jones, F.B., Concerning normality and completely normal spaces, Bull. amer. math. soc., 43, 671-676, (1937) · JFM 63.1171.03
[14] Junnila, H.J.K, Countability of point finite families of sets, Canad. J. math., 31, 673-679, (1979) · Zbl 0347.28002
[15] Matveev, M.V., On properties similar to pseudocompactness and countable compactness, Moscow univ. math. bull., 39, 32-36, (1984) · Zbl 0556.54016
[16] McIntyre, D.W., Chain conditions in linearly ordered and regular first countable spaces, () · Zbl 0746.54002
[17] Moore, R.L., Foundations of point set theory, American mathematical society colloquim publications, 13, (1962), Amer. Math. Soc Providence, RI, rev. ed. · Zbl 0192.28901
[18] Mrówka, S., On completely regular spaces, Fund. math., 41, 105-106, (1954) · Zbl 0055.41304
[19] Pixley, C.; Roy, P., Uncompletable Moore spaces, Proceedings auburn university topology conference, 75-85, (1969) · Zbl 0259.54022
[20] Reed, G.M., Concerning normality, metrizability, and the souslin property in subspaces of Moore spaces, General topology appl., 1, 223-246, (1971) · Zbl 0224.54040
[21] Reed, G.M., On chain conditions in Moore spaces, General topology appl., 4, 255-267, (1974) · Zbl 0295.54042
[22] Reed, G.M., On continuous images of Moore spaces, Canad. J. math., 26, 1475-1479, (1974) · Zbl 0312.54031
[23] Sarkhel, D.N., Some generalizations of countable compactness, Indian J. pure appl. math., 17, 778-785, (1986) · Zbl 0628.54017
[24] Scott, B.M., Pseudocompact, metacompact spaces are compact, Topology proc., 4, 577-587, (1979) · Zbl 0449.54020
[25] Wiscamb, M.R., The discrete countable chain condition, Proc. amer. math. soc., 23, 608-612, (1969) · Zbl 0184.26304
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