Possible point-open types of subsets of the reals.

*(English)*Zbl 0719.54003Summary: 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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##### References:

[1] | Daniels, P.; Gruenhage, G., The point-open type of subsets of the reals, Topology appl., 37, 53-64, (1990) · Zbl 0718.54015 |

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