## 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
Full Text:

### References:

 [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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.