# On <i>ψ</i> (<i>κ , Μ</i>) spaces with <i>κ</i> = <i>ω</i><sub>1</sub>.

- 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* = *ω*_{1} 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 *ω*_{1} such that *Μ* has cardinality **c** and a continuous function *f* : *ψ* (*ω*_{1} , *Μ*) *ω* → [0,1] such that for every *r* ∈ [0,1], ⃒*f* ^{-1}(*r*)⃒ < **c** = ⃒*Μ*⃒. Dow and Vaughan proved that **α** < **c** is
equivalent to a similar statement with ω in the place of ω_{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 *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*)⃒ ∩ *Μ*⃒ is at least α. Finally, we will provide some generalizations and interpretations of related lemmas in the *ω*_{1} case.

On <i>ψ</i> (<i>κ , Μ</i>) spaces with <i>κ</i> = <i>ω</i><sub>1</sub>.

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Created on 5/1/2010

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