On [psi (kappa, Mu)] spaces with [kappa equals omega subscript 1]

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. Mrowka introduced a topological space [psi] 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 [psi (kappa, Mu)] spaces based on any cardinal [kappa] together with a MADF of countably infinite subsets of [kappa]. They proved new results, including new results for the case [kappa equals omega]. 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 [kappa equals omega] to the [kappa equals omega subscript 1] case. Our goal was to show that the cardinal inequality [a is less than c], where [a] is the smallest cardinality of a MADF on [omega], is equivalent to the condition that there exists a MADF [Mu] of infinite subsets of [omega subscript 1] such that [Mu] has cardinality [c] and a continuous function [f] : [psi(omega subscript 1, Mu) then [0,1]] such that for every [r] [that is a member of the set] [0,1], |[f superscript -1(r)]| < [c equals |Mu|]. Dow and Vaughan proved that [a is less than c] is equivalent to a similar statement with [omega] in the place of [omega subscript 1], 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 [Mu] and function [f] implies that [a is less than c]. One important result that we show along the way to our main result is that for any continuous function from [psi (kappa, Mu)] into the interval [0,1], there is some [r] [that is a member of the set] [0,1] such that |[f superscript -1(r) intersects Mu]| is at least [a]. Finally, we will provide some generalizations and interpretations of related lemmas in the [omega subscript 1] case.

Additional Information

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

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