Abstract
AbstractWe introduce natural strengthenings of sequential compactness, the r-Ramsey property for each natural number $r\geq 1$ . We prove that metrizable compact spaces are r-Ramsey for all r and give examples of compact spaces that are r-Ramsey but not $(r+1)$ -Ramsey for each $r\geq 1$ (assuming Continuum Hypothesis (CH) for all $r>1$ ). Productivity of the r-Ramsey property is considered.
Highlights
Let K be a compact space, and let r be a positive integer
Following [2], we say that a function f ∶ [S]r → K converges to p ∈ K if for every neighborhood U of p, there is a finite set F such that f ([S/F]r) ⊆ U
We show that a more general notion of a space satisfying the r-Ramsey property (Definition 2.1) given below is a quite natural strengthening of sequential compactness, and the main motivation of this paper is to prove some basic facts about this class of spaces and describe some examples showing that r-Ramsey can be strictly weaker than (r + 1)Ramsey
Summary
Following [2], we say that a function f ∶ [S]r → K converges to p ∈ K if for every neighborhood U of p, there is a finite set F such that f ([S/F]r) ⊆ U. Once this happens for some p, we say that f is convergent This notion, for r = 2, was introduced in [2] where the special case of our Theorem 2.1 was stated and proved. Their main motivation was to obtain idempotents in compact semigroups K as limits of certain functions f ∶ [ω]2 → K. Set-theoretic notation and terminology, including some background on Ramsey’s theorem, can be found in [5], and for a more detailed analysis of almost disjoint families, Ψ spaces, and the ideals FINn, we refer the reader to [4]
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