One of the most basic and natural generalizations of first countability is countable tightness: the condition that, whenever^ is in the closure of A, there is a countable subset B of A such that x 6 B. Countably tight spaces include sequential spaces, i.e., those in which closure is obtainable by iteration of the operation of taking limits of convergent sequences. The two classes are distinct, since there are easy examples of countable, nondiscrete spaces with only trivial convergent sequences. On the other hand, it was long a major unsolved problem whether every compact Hausdorff space of countable tightness is sequential. First posed in [1] and motivated by the main results of [2], it gained importance from subsequent discoveries on the strong structural properties enjoyed by compact sequential spaces (see [3] and its references). A negative answer was shown to be consistent by Ostaszewski [4], who used Gödel's Axiom of Constructibility (V = L) to construct a countably compact space X whose one-point compactification is countably tight; of course, no sequence from X can converge to the extra point. Now (Theorem 2) we have shown that a positive answer follows from the Proper Forcing Axiom (PFA), introduced in [5]. Our research has uncovered many other striking consequences of PFA, numbered below. None was known to be consistent until now, nor was the following consequence of Corollary 1 and Theorem 3: the ordinal space oj\ embeds in every first countable, countably compact, noncompact Ti space (in particular, in every countably compact nonmetrizable T
Abstract. We investigate some combinatorial statements that are strong enough to imply that ♦ fails (hence the name antidiamonds); yet most of them are also compatible with CH. We prove that these axioms have many consequences in set-theoretic topology, including the consistency, modulo large cardinals, of a Yes answer to a problem on linearly Lindelöf spaces posed by Arhangel'skiȋ and Buzyakova (1998).
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