Countable tightness and proper forcing

Abstract

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<i manifold), assuming PFA. This remains true if "first countable" is weakened to "character < ui," meaning every point has a local base of cardinality < u)i. This leads to a remarkable structure theorem for regular, countably compact spaces of character < ui (e.g. the product space [0, l] Wl ): under PFA, the closure of every set A can be taken by first adjoining all limits of convergent sequences, and then adjoining to the resulting set A" all points x for which there is a copy W of w\ in A~ such that W U {x} is homeomorphic to ui + 1.Although large cardinals are needed to prove the consistency of PFA, all our PFA results are consistent if ZF is consistent. This is established by using u^-p-i-C-[6, Chapter VIII] posets and a ground model with a o W2 -sequence to capture approximations to possible counterexamples in a countable support

On complete accumulation points of discrete subsets

“…Both of the examples we have just mentioned are S-spaces (regular, hereditarily separable but not hereditarily Lindelöf) and it is well-known that the existence of an S-space is independent of ZF C. We also note that the existence of a discretely complete, non-compact regular space of countable spread is independent of ZF C since it was shown in [2] that under the Proper Forcing Axiom each regular countably compact space of countable spread is compact. Furthermore, Peter Nyikos has informed us that the same result holds in the case of countably compact Hausdorff spaces.…”

First countable, countably compact spaces and the continuum hypothesis

“…In this section, we study the structure of first countable topological spaces that can be mapped by a closed continuous function onto the space ω 1 . In particular, we show that such spaces have much in common with ω 1 -one can define analogs of "closed unbounded" and "stationary".…”

“…This natural example brings to mind the question of to what extent must a noncompact first countable, countably compact space resemble ω 1 . A more precise formulation of this is "is it true that every first countable, countably compact space is either compact, or contains a closed subset homeomorphic to ω 1 ?".…”

“…The goal of this paper is to show that the Continuum Hypothesis does not decide the answer to the question -we produce a model of ZFC in which the Continuum Hypothesis holds and in which first countable, countably compact spaces are either compact or contain a closed subset homeomorphic to ω 1 . The results and method contained in this paper have had some impact on current research in set-theoretic topology.…”

“…In section 2, we show that first countable closed pre-images of ω 1 of size ω 1 behave a lot like ω 1 itself -there are natural analogs of the closed unbounded filter and a lot of the properties of the club filter carry over into this more general context. In section 3, we define a notion of forcing that shoots a copy of ω 1 through a given first countable closed pre-image of ω 1 . In sections 4 and 5 we define weakly < ω 1 -proper notions of forcing, verify that the notion of forcing defined in section 3 is weakly < ω 1 -proper, and prove that this property is preserved by countable support iterations.…”