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Combinatorics of open covers (III): games, Cp (X)

Marion Scheepers — 1997

Fundamenta Mathematicae

Some of the covering properties of spaces as defined in Parts I and II are here characterized by games. These results, applied to function spaces C p ( X ) of countable tightness, give new characterizations of countable fan tightness and countable strong fan tightness. In particular, each of these properties is characterized by a Ramseyan theorem.

Lindelöf indestructibility, topological games and selection principles

Marion ScheepersFranklin D. Tall — 2010

Fundamenta Mathematicae

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most 2 . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are G δ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property,...

Combinatorics of open covers (VII): Groupability

Ljubiša D. R. KočinacMarion Scheepers — 2003

Fundamenta Mathematicae

We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a T 31 / 2 -space. In [9] we showed that C p ( X ) has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. C p ( X ) has countable fan tightness and the Reznichenko property. 2....

Gδ -sets in topological spaces and games

Winfried JustMarion ScheepersJuris StepransPaul Szeptycki — 1997

Fundamenta Mathematicae

Players ONE and TWO play the following game: In the nth inning ONE chooses a set O n from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset T n of X. The players must obey the rule that O n O n + 1 T n + 1 T n for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a G δ -set. To what extent is the converse true? We show that:  (A) For ℱ the collection of countable subsets of X:   1. There are subsets...

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