How Much Weak Compactness Does the Weakly Compact Reflection Principle Imply?
نویسنده
چکیده
The weakly compact reflection principle Reflwcpκq states that κ is a weakly compact cardinal and every weakly compact subset of κ has a weakly compact proper initial segment. The weakly compact reflection principle at κ implies that κ is an ω-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that κ is pω ` 1qweakly compact. Moreover, we show that if the weakly compact reflection principle holds at κ then there is a forcing extension preserving this in which κ is the least ω-weakly compact cardinal. Along the way we generalize the wellknown result which states that if κ is a regular cardinal then in any forcing extension by κ-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if κ is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length κ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.
منابع مشابه
Dependent Choices and Weak Compactness
We work in set-theory without the Axiom of Choice ZF. We prove that the principle of Dependent Choices (DC) implies that the closed unit ball of a uniformly convex Banach space is weakly compact, and in particular, that the closed unit ball of a Hilbert space is weakly compact. These statements are not provable in ZF, and the latter statement does not imply DC. Furthermore, DC does not imply th...
متن کاملRepresenting Completely Continuous Operators through Weakly ∞-compact Operators
Let V,W∞, andW be operator ideals of completely continuous, weakly ∞-compact, and weakly compact operators, respectively. We prove that V =W∞ ◦W−1. As an immediate application, the recent result by Dowling, Freeman, Lennard, Odell, Randrianantoanina, and Turett follows: the weak Grothendieck compactness principle holds only in Schur spaces.
متن کاملA Weak Grothendieck Compactness Principle
The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in...
متن کاملSEQUENTIAL S ∗ - COMPACTNESS IN L - TOPOLOGICAL SPACES ∗ SHU - PING LI Mudanjiang
In this paper, a new notion of sequential compactness is introduced in L-topological spaces, which is called sequentially S∗-compactness. If L = [0, 1], sequential ultra-compactness, sequential N-compactness and sequential strong compactness imply sequential S∗-compactness, and sequential S∗-compactness implies sequential F-compactness. The intersection of a sequentially S∗-compact L-set and a ...
متن کامل