On ψ (κ , Μ) spaces with κ = ω₁.

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)

Catherine Ann Payne (Creator)

Institution

The University of North Carolina at Greensboro (UNCG )

Web Site: http://library.uncg.edu/

Advisor

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.

On ψ (κ , Μ) spaces with κ = ω₁.
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Created on 5/1/2010
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Additional Information

Publication

Thesis

Language: English

Date: 2010

Keywords

Disjoint, maximal, mrowka, topology

Subjects

Topology.

Cardinal numbers.

Maximal functions.