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 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.
In [Fund. Math. 210 (2010), 1-46] we claimed the truth of two statements, one now known to be false and a second lacking a proof. In this "Errata" we report these matters in the interest of setting the record straight on the status of these claims.
Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are 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,...
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 -space. In [9] we showed that 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. has countable fan tightness and the Reznichenko property. 2....
Players ONE and TWO play the following game: In the nth inning ONE chooses a set from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset of X. The players must obey the rule that 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 -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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