Atlas: Applications of independent families by Wanjun Hu

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SumTopo 2001, Sixteenth Summer Conference on Topology and its Applications
July 18-21, 2001
City College of CUNY
New York, NY, USA

Organizers
Ralph Kopperman (City College, CUNY), Susan Andima (CW Post College, LIU), Gerald Itzkowitz (Queens College, CUNY), Prabudh Misra (College of Staten Island, CUNY), Shelly Rothman (CW Post College, LIU), Aaron Todd (Baruch College, CUNY)

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Applications of independent families
by
Wanjun Hu
Wesleyan Univ.
Coauthors: W.W. Comfort

Let I be a family of subsets of X. It is called a (\theta, \kappa)-independent family if every family of size < \theta of sets from I or the complements of elements in I has an intersection of size at least \kappa ([JECH], [HAUS]).

Independent families have been used in several places([E], [VAN]) to construct maximal Tychnoff topologies. We shall show that independent families introduce tychnoff topologies naturally, and that kind of topologies is in fact related to dense subsets of Cantor cubes. We prove that for any cardinal \kappa, there exists a maximal independent family of size 2^\kappa that is also maximal \kappa-independent. This answers a question by Eckertson([E]). Based on these results and some other results, a joint work with Dr. W.W. Comfort has been submitted.

A generalization of independent families has also be discussed. Using that generalization, we can provide examples of tychnoff spaces that are not maximally resolvable, concerning the question of the existence of a countable resolvable but not maximally resolvable tychnoff space(See [CG]). We also prove some results of equivalence between the existence of maximal generalized independent families and that of usual independent families. Note also that Kunen in [KUN] proved that the existence of a maximal (\omega₁, \kappa)-independent family is equiconsistent with a measurable cardinal.

Reference:

[CG] W.W. Comfort and S. Garcia-Ferreira, Resolvability: a selective survey and some new results,

[E] F.W. Eckertson, Resolvable, not maximally resolvable spaces, Top. Appl. 79(1997)1-11

[VAN] E.K. van Douwen, Applications of maximal topologies, Top. Appl. 51(1993)125-139

[HAUS] F. Hausdorff, Über zwei Sätze von G. Fichtenholz und L. Kantorovitch, Studia Math. 6(1936), pp. 18-19.

[HEW] E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10(1943) 309-333

[JECH] T. Jech, Set theory, second edition, Springer-verlag, 1997.

[KUN] K. Kunen, Maximal \sigma-independent families, Fund. Math. 117(1983)75-80.

[KST] K. Kunen, A. Szymanski and F. Tall, Baire resolvable spaces and ideal theory, Prace Nauk., Ann. Math. Sil. 2(14)(1986)98-107.

http://at.yorku.ca/i/d/e/c/27.htm

Date received: May 16, 2001