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Combinatorics of open covers. VII: Groupability. (English) Zbl 1115.91013

Combinatorial properties of open covers are studied. Ramseyan partition relations are used to characterize: (1) the classical covering property of Hurewicz, (2) the covering property of Gerlits and Nagy, and (3) the combinatorial cardinal numbers \(b\) and \(\text{add}(M)\). Let \(X\) be a Tikhonov space and \(C_p(X)\) the corresponding space of real-valued continuous functions on \(X\), endowed with the topology of pointwise convergence. It is shown that the following are equivalent: (1) \(C_p(X)\) has countable fan tightness and the Reznichenko property and (2) all finite powers of \(X\) have the Hurewicz property.
Part VI, cf. the second author, Quaest. Math. 22, No. 1, 109–130 (1999; Zbl 0972.91026). More recent parts have appeared (VIII: Topology Appl. 140, No. 1, 15–32 (2004; Zbl 1051.54019), IX: Note Mat. 22 (2003–2004), No. 2, 167–178 (2004; Zbl 1115.54009), XI: Topology Appl. 154, No. 7, 1269–1280 (2007; Zbl 1114.54023)).

MSC:

91A44 Games involving topology, set theory, or logic
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
03E02 Partition relations
05D10 Ramsey theory
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54C65 Selections in general topology
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