Pseudoradial spaces and copies of ω1 + 1 Bella, Angelo; Dow, Alan; Hernández-Gutiérrez, Rodrigo
Topology and its applications,
03/2020, Volume:
272
Journal Article
Peer reviewed
Open access
In this paper we compare the concepts of pseudoradial spaces and the recently defined strongly pseudoradial spaces in the realm of compact spaces. We show that MA+c=ω2 implies that there is a compact ...pseudoradial space that is not strongly pseudoradial. We essentially construct a compact, sequentially compact space X and a continuous function f:X→ω1+1 in such a way that there is no copy of ω1+1 in X that maps cofinally under f. We also give some conditions that imply the existence of copies of ω1 in spaces. In particular, PFA implies that compact almost radial spaces of radial character ω1 contain many copies of ω1.
We show that there are models of MAω1 where the Σ31-uniformization property holds. Further we show that “BPFA+ ℵ1 is not inaccessible to reals” outright implies that the Σ31-uniformization property ...is true.
We answer questions about P-filters in the Cohen, random, and Laver forcing extensions of models of CH. In the case of the ℵ2-random real poset, we prove that if □ℵ1 also holds in the ground model, ...then there are P-points of ω⁎ in the extension. The majority of the paper investigates the question of whether ω⁎ can be covered by nowhere dense P-sets. We prove that this is not the case if ℵ2-Cohen reals are added to a model of CH in which □ω1 holds, and that it is the case in the standard Laver extension. We also answer questions formulated by P. Nyikos about interactions between ultrafilter orderings of ωω and mod finite scales. We show they have connections to ultrafilters having non-meager P-subfilters.
Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is
ℵ
2
. The Mapping Reflection Principle follows from the ...Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from the Proper Forcing Axiom, for example, the
P
-ideal Dichotomy. In this article, it is proved that the Y-Proper Forcing Axiom implies the Mapping Reflection Principle by introducing forcing notions whose conditions are finite objects.
Between reduced powers and ultrapowers, II Farah, Ilijas; Shelah, Saharon
Transactions of the American Mathematical Society,
December 1, 2022, 2022-12-00, Volume:
375, Issue:
12
Journal Article
Peer reviewed
Open access
We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) ...structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.
We prove that the Hurewicz property is not preserved by finite products in the Miller model. This is a consequence of the fact that Miller forcing preserves ground model \gamma -spaces.
P. J. Nyikos has asked whether it is consistent that every hereditarily normal manifold of dimension greater than one is metrizable, and he proved that it is if one assumes the consistency of a ...supercompact cardinal, and, in addition, that the manifolds are hereditarily collectionwise Hausdorff. We are able to omit these extra assumptions.
In this article, we consider the notion of almost irredundant sets: A subset X of a C*-algebra A is called almost irredundant if and only if for every a∈X, the element a does not belong to the ...norm-closure of{∑i=1nλi∏j=1niai,j:whereai,j∈X∖{a}and∑|λi|≤1}. Since every almost irredundant set is in particular a discrete set, it follows that the density of A is an upper bound for the size of almost irredundant sets. We prove that under the Proper Forcing Axiom (PFA), there is an uncountable almost irredundant set in every C*-algebra with an uncountable increasing sequence of ideals. In particular, assuming PFA, every nonseparable scattered C*-algebra admits an uncountable almost irredundant set.
Definable MAD families and forcing axioms Fischer, Vera; Schrittesser, David; Weinert, Thilo
Annals of pure and applied logic,
20/May , Volume:
172, Issue:
5
Journal Article
Peer reviewed
Open access
We show that ZFC + BPFA (i.e., the Bounded Proper Forcing Axiom) + “there are no Π21 infinite MAD families” implies that ω1 is a remarkable cardinal in L. In other words, under BPFA and an anti-large ...cardinal assumption there is a Π21 infinite MAD family. It follows that the consistency strength of ZFC + BPFA + “there are no projective infinite MAD families” is exactly a Σ1-reflecting cardinal above a remarkable cardinal. In contrast, if every real has a sharp—and thus under BMM—there are no Σ31 infinite MAD families.
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