The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regula...

The Noetherian type of a space is the least κ for which the space has a κop-like base, i.e., a base in which no element has κ-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products thereof. For example: the density of a product of not-too-many compact linear orders never exceeds its Noetherian type, with equality possible only for singular Noet...

We show that it is consistent relative to a weakly compact cardinal strong homology additive and compactly supported within the class of locally separable metric spaces. This complements work Mardešić Prasolov [14] showing Continuum Hypothesis implies countable sum Hawaiian earrings witnesses failure possess either these properties. Our results build directly on Lambie-Hanson second author [3] ...

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ?-based constructions Souslin trees with various additional properties may be rendered as applications our approach. paper, show following same carried out even in absence ?. particular, obtain a new weak sufficient condition for existence at level strongly inaccessible cardi...

In the context of large cardinals, the classical diamond principle 3κ is easily strengthened in natural ways. When κ is a measurable cardinal, for example, one might ask that a 3κ sequence anticipate every subset of κ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function ` ... κ → Vκ such that for any A ∈ H(κ) there is an embedding...

We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of Lκω and Lκκ: For weakly compact κ there is no strongest extension of Lκω with the (κ, κ)compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of Lκκ with the (κ, κ)-compactness property and the LöwenheimS...

in this paper, a complete description concerning linear operators of banach spaces with range in lipschitz algebras $lip_al(x)$ is provided. necessary and sufficient conditions are established to ensure boundedness and (weak) compactness of these operators. finally, a lower bound for the essential norm of such operators is obtained.

By introducing the concepts of order almost Dunford-Pettis and almost weakly limited operators in Banach lattices, we give some properties of them related to some well known classes of operators, such as, order weakly compact, order Dunford-Pettis, weak and almost Dunford- Pettis and weakly limited operators. Then, we characterize Banach lat- tices E and F on which each operator from E into F t...

It is proved that for every uncountable cardinal λ, GCH+ (λ) entails the existence of a cf(λ)-complete λ-Souslin tree. In particular, if GCH holds and there are no א2-Souslin trees, then א2 is weakly compact in Gödel’s constructible universe, improving Gregory’s 1976 lower bound. Furthermore, it follows that if GCH holds and there are no א2 and א3 Souslin trees, then the Axiom of Determinacy ho...

It is proved, under Martin’s Axiom, that all (ω1, ω1) gaps in P(N) / fin are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin’s Axiom the Baire categoricity of all ∆3 non-∆ 1 3-complete sets ...