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Reviewed by Klaas Pieter Hart References

By Alan S. (-nc-ms) Vaughan and Jerry E. (-ncg-ms

Abstract

Ordinal remainders of classical ψ-spaces. (English summary) Fund. Math. 217 (2012), no. 1, 83–93.1730-6329 The ‘classical ’ ψ-spaces alluded to in the title are those built from (maximal) almost disjoint families on ω. Given such a family, A, say, one topologises ω ∪ A by declaring each point of ω to be isolated and having A ∪ A be clopen and homeomorphic with a convergent sequence, with the point A as the limit—the resulting space is denoted ψ(ω, A) or, if A is agreed upon, simply ψ. These spaces have a rich history, as (counter)examples and as objects of independent study. The authors prove that if λ is an ordinal and there is a chain of order type λ of infinite subsets of ω, with respect to the order ⊂ ∗ of containment mod-finite, then there is a maximal almost disjoint family A such that βψ � ψ is homeomorphic to the ordinal λ + 1. This is complemented by exhibiting such chains of any order type less than t +, where t is the familiar tower number

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.372.3102
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