Mrówka maximal almost disjoint families for uncountable cardinals

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• 作者：Alan Dow ; Jerry E. Vaughan
• 关键词：Almost disjoint families ; Mr&#xf3 ; wka– ; Isbell Ψ ; -spaces ; Continuous real-valued functions ; Cardinal numbers ; Countable cofinality ; Stone– ; Č ; ech compactification
• 刊名：Topology and its Applications
• 出版年：2010
• 出版时间：1 June 2010
• 年：2010
• 卷：157
• 期：8
• 页码：1379-1394
• 全文大小：336 K

文摘

We consider generalizations of a well-known class of spaces, called by S. Mrówka, , where is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers N. We denote these generalizations by for κω. Mrówka proved the interesting theorem that there exists an such that . In other words there is a unique free z-ultrafilter p₀ on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for , Mrówka's MADF can be used to produce a MADF such that . For , and every , it is always the case that , yet there exists a special free z-ultrafilter p on retaining some of the properties of p₀. In particular both p and p₀ have a clopen local base in βψ (although need not be zero-dimensional). A result for , that does not apply to p₀, is that for certain , p is a P-point in βψ.