The linear refinement number and selection theory

DOI10.4064/FM124-8-2015arXiv1404.2239MaRDI QIDQ6250580

Boaz Tsaban, Saharon Shelah, Michał Machura

Publication date: 8 April 2014

Abstract: The emph{linear refinement number}

m a t h f r a k l r

is the minimal cardinality of a centered family in

[o m e g a]^{o} m e g a

such that no linearly ordered set in

([o m e g a]^{o} m e g a, s u b s e t e q^{*})

refines this family. The emph{linear excluded middle number}

m a t h f r a k l x

is a variation of

m a t h f r a k l r

. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classic combinatorial cardinal characteristics of the continuum. We prove that

m a t h f r a k l r = m a t h f r a k l x = m a t h f r a k f d

in all models where the continuum is at most

a l e p h_{2}

, and that the cofinality of

m a t h f r a k l r

is uncountable. Using the method of forcing, we show that

m a t h f r a k l r

and

m a t h f r a k l x

are not provably equal to

m a t h f r a k d

, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

Mathematics Subject Classification ID

Selections in general topology (54C65) Consistency and independence results (03E35) Cardinal characteristics of the continuum (03E17)

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