On <i>&#x03C8</i> (<i>&#x03BA , &#x039C</i>) spaces with <i>&#x03BA</i> = <i>&#x03C9</i><sub>1</sub>.

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Catherine Ann Payne (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
Jerry Vaughan

Abstract: S. Mrὀwka introduced a topological space ψ whose underlying set is the natural numbers together with an infinite maximal almost disjoint family(MADF) of infinite subsets of natural numbers. A. Dow and J. Vaughan proved a number of results for similar ψ (κ , Μ) spaces based on any cardinal κ together with a MADF of countably infinite subsets of κ. They proved new results, including new results for the case κ = ω. In this paper, we will review some properties of the spaces ψ (κ , Μ) for any cardinal κ. We will then extend some of the results of Dow and Vaughan for κ = ω to the κ = ω1 case. Our goal was to show that the cardinal inequality α < c, where α is the smallest cardinality of a MADF on ω, is equivalent to the condition that there exists a MADF Μ of infinite subsets of ω1 such that Μ has cardinality c and a continuous function f : ψ (ω1 , Μ) ω → [0,1] such that for every r ∈ [0,1], ⃒f -1(r)⃒ < c = ⃒Μ⃒. Dow and Vaughan proved that α < c is equivalent to a similar statement with ω in the place of ω1, and although we were able to generalize some of the relevant lemmas, at this time we are only able to prove that the existence of such a MADF Μ and function f implies that α < c. One important result that we show along the way to our main result is that for any continuous function from ψ (κ , Μ) into the interval [0,1], there is some r ∈ [0,1] such that ⃒f -1(r)⃒ ∩ Μ⃒ is at least α. Finally, we will provide some generalizations and interpretations of related lemmas in the ω1 case.

Additional Information

Language: English
Date: 2010
Disjoint, maximal, mrowka, topology
Cardinal numbers.
Maximal functions.

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