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Possible point-open types of subsets of the reals. (English) Zbl 0719.54003
Summary: If X is a topological space and \(\alpha\) is an ordinal, then the point- open game of length \(\alpha\) on X, abbreviated \(G_{\alpha}(X)\), is the two person game of length \(\alpha\) in which, on the \(\beta\) th move \((\beta <\alpha)\), the first player (the “point picker”) picks a point of X and the second player (the “open set picker”) picks an open subset of X covering the point just played. The point picker wins iff the open sets thus picked cover X. The point-open type of X, abbreviated pot(X), is defined to be the smallest ordinal \(\alpha\) such that the point picker has a winning strategy in \(G_{\alpha}(X)\). This ordinal clearly exists and is no more than the cardinality of X. The main result of this paper is that if we assume the continuum hypothesis, then for every limit ordinal \(\alpha <\omega_ 1\), there is a subset X of the real numbers such that \(pot(X)=\alpha\). This solves a problem due to P. Daniels and G. Gruenhage [ibid. 37, No.1, 53-64 (1990; Zbl 0718.54018)].

MSC:
54A35 Consistency and independence results in general topology
03E50 Continuum hypothesis and Martin’s axiom
91A44 Games involving topology, set theory, or logic
03E15 Descriptive set theory
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