Cardinal sequences and Cohen real extensions

Abstract

Abstract. We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most (2 ℵ 0 ) V levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.

Compact scattered spaces in forcing extensions

“…Similarly to [3], our argument assumes that (B) is false, and produces an infinite independent sequence of clopen sets, contradicting the assumption that X is scattered. Our plan is to associate trees with scattered spaces, and prove (Lemma 6) that ¬(B) will yield a tree of large rank, whereas (Lemma 9) in certain forcing extensions, all such trees must have small rank.…”

“…Just's model was the standard Cohen real extension over a model of CH. This result was improved by [3], which showed that (B) holds in the Cohen extension. The argument in [3] is very specific to Cohen forcing, and…”

“…We show here (see Theorem 8) that in fact (B) does hold in this model, and also in the model obtained by adding ℵ 2 random reals in the standard way (by one measure algebra). We do not know whether (B) follows from some abstract principle, such as C s (ω 2 ) described by Juhász, Soukup, and Szentmiklóssy [4]; this is also asked in [3]. However, since (B) is true in the random real model, where C s (ω 2 ) is false (because there is an ω 2 -Luzin gap), it would be more interesting if one could derive (B) from some abstract principle true in both the Cohen and random extensions.…”

“…The argument in [3] is very specific to Cohen forcing, and they ask whether (B) holds in similar extensions, such as the one obtained by adding ℵ 2 random reals side-by-side. We show here (see Theorem 8) that in fact (B) does hold in this model, and also in the model obtained by adding ℵ 2 random reals in the standard way (by one measure algebra).…”

“…Some remarks: [3] states (B) for locally compact spaces, but that is equivalent (by taking the one-point compactification). Likewise, if (A) holds, one can remove the top level and get a locally compact X of height ω 2 such that all I(X (α) ) are countable.…”