Absolutely countably compact spaces. (English) Zbl 0801.54021

Matveev, Michael V.

Topology Appl. 58, No. 1, 81-92 (1994).

This paper introduces an interesting new star covering property. A space \(X\) is called absolutely countably compact (acc) provided for every open cover \({\mathcal U}\) of \(X\) and every dense \(Y\subset X\), there exists a finite set \(F\subset Y\) such that \(\bigcup \{U\in {\mathcal U}\): \(U\cap F \neq\emptyset\} =X\). Every compact space is acc, and every acc \(T_ 2\)- space is countably compact. It is, therefore, somewhat unexpected that the acc property is not necessarily preserved by continuous mappings, products with compact spaces, or passage to closed subsets (indeed every countably compact space can be embedded as a closed subset of some acc space). Every countably compact space of countable tightness is acc, as is every \(\Sigma\)-product of compact spaces of countable tightness. If \(X\) is a \(T_ 1\) non-compact space, then there exists a cardinal \(\tau\) such that the product \(X^ \tau\) is not acc. A space \(X\) is called hacc if every closed subset of \(X\) is acc. The product of a \(T_ 2\) hacc (acc) space with a compact first countable space is hacc (acc). Several other related star covering properties are briefly considered.

Reviewer: J.E.Vaughan (Greensboro)

MSC:

54D30	Compactness
54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)
54B10	Product spaces in general topology

Keywords:

star covering property; countably compact space; \(\Sigma\)-product of compact spaces; countable tightness

References:

[1]	Engelking, R., General topology, (1977), PWN Warsaw
[2]	Fleishman, W.M., A new extension of countably compactness, Fund. math., 67, 1-9, (1970)
[3]	Franklin, S.P.; Rajagopalan, M., Some examples in topology, Trans. amer. math. soc., 155, 305-314, (1971) · Zbl 0217.48104
[4]	Ikenaga, S., Some properties of ω-n-star spaces, Res. rep. Nara nat. college tech., 23, 53-57, (1987)
[5]	Kombarov, A.P.; Malyhin, V.I., On σ-products, Dokl. akad. nauk USSR, 213, 774-776, (1973), (in Russian); also: (in English) · Zbl 0296.54008
[6]	Matveev, M.V., On properties similar to pseudocompactness and countably compactness, Vestnik MGU ser. 1, Moscow univ. math. bull., 39, 32-36, (1984), also: (in English) · Zbl 0556.54016
[7]	Novak, J., On the Cartesian product of two compact spaces, Fund. math., 40, 106-112, (1953) · Zbl 0053.12404
[8]	Niyikos, P., On first countable, countably compact spaces III: the problem of obtaining separable noncompact examples, (), 127-161
[9]	Stephenson, R.M., Initially k-compact and related spaces, (), 603-632
[10]	Tkačuk, V.V., Remainders of discrete spaces — some applications, Vestnik MGU ser. 1, 4, 18-21, (1990), (in Russian); also: (in English)
[11]	van Douwen, E.K.; Reed, G.M.; Roscow, A.W.; Tree, I.J., Star covering properties, Topology appl., 39, 71-103, (1991) · Zbl 0743.54007
[12]	Vaughan, J.E., Countably compact and sequentially compact spaces, (), 569-602 · Zbl 0383.54013