The combinatorics of splittability

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Abstract

Marion Scheepers, in his studies of the combinatorics of open covers, introduced the property Split(U,V) asserting that a cover of type U can be split into two covers of type V. In the first part of this paper we give an almost complete classification of all properties of this form where U and V are significant families of covers which appear in the literature (namely, large covers, ω-covers, τ-covers, and γ-covers), using combinatorial characterizations of these properties in terms related to ultrafilters on ℕ. In the second part of the paper we consider the questions whether, given U and V, the property Split(U,V) is preserved under taking finite or countable unions, arbitrary subsets, powers or products. Several interesting problems remain open.

Original languageEnglish
Pages (from-to)107-130
Number of pages24
JournalAnnals of Pure and Applied Logic
Volume129
Issue number1-3
DOIs
StatePublished - Oct 2004
Externally publishedYes

Bibliographical note

Funding Information:
Partially supported by the Golda Meir Fund and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). E-mail address: tsaban@math.huji.ac.il (B. Tsaban).

Keywords

  • Hereditarity
  • P-point
  • Powers
  • Products
  • Splitting
  • Ultrafilter
  • γ-Cover
  • τ-Cover
  • ω-Cover

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