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Remarks on the numbers of ideals of Boolean algebra and open sets of a topology

Part of the Lecture Notes in Mathematics book series (LNM,volume 1182)

Abstract

We prove that the cardinals μ which may be the number of ideals of an infinite Boolean algebras are restricted: \(\mu = \mu ^{\aleph _0 }\) and if κ≤μ is strong limit then μ<κ=μ. Similar results hold for the number of open sets of a compact space (we need w(x) <ŝ(x)=2<ŝ(x)). We also prove that if μ≥⊃2 is the number of open subsets of a Hausdorff space X,\(\mu < \mu ^{\aleph _0 }\) then 0# exists, (in fact, the consequences of the covering lemma on cardinal arithmetic are violated). We also prove that if the spread μ of a Hausdorff space X satisfies μ>⊃2(c f μ) that the sup is obtained. For regular spaces μ;>2cf μ is enough.

Similarly for 3(X) and h (X).

Keywords

  • Open Subset
  • Topological Space
  • Boolean Algebra
  • Strong Limit
  • Hausdorff Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. A. Hajnal and I. Juhasz, Some remarks on a property of topological cardinal functions, Acta Math. Acad. Sci. Hungar, 20 (1969), 25–37.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. A. Hajnal and I. Juhasz, On the number of open sets, Ann. Univ. Sci. Budapest. 16 (1973), 99–102.

    MathSciNet  MATH  Google Scholar 

  3. I. Juhasz, Cardinal functions in topology, Math. Center Tracts. Amsterdam, 1971.

    Google Scholar 

  4. I. Juhasz, Cardinal functions in topology-ten year later. Math. Center Tracts. Amsterdan, 1980.

    Google Scholar 

  5. I. Juhasz and S. Shelah, How large can a hereditary separable or hereditarily Lindel of space by? Israel J. of Math, submitted.

    Google Scholar 

  6. K. Kunen and J. Roitman, Attaining the spread of cardinals of cofinality ω, Pacific J. Math. 70 (1977), 199–205.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. J. Roitman, Attaining the spread at cardinals which are not strong limit, Pacific J. Math. 57 (1975), 545–551.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. S. Shelah, Remarks on Boolean algebra, Algebra Universalis, 11 (1980), 77–89.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. _____, Canonization theorems and applications, J. of Symb. Logic. 46 (1981), 345–353.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. _____, On cardinal invariants in topology, General Topology and its applications, 7 (1977), 251–259.

    MathSciNet  MATH  Google Scholar 

  11. _____, On some problem in general topology, a preprint, Jan. 1978.

    Google Scholar 

  12. _____, If ◊ℵ1 + "there is an ℵ1-Kurepa tree with κ-branches" then some B.A. of power ℵ1 has λ filters and λℵ0-ultrafilters. Mimeograph Notes from Madison, Fall 77.

    Google Scholar 

  13. _____S. Shelah, On P-points, β(ω) and other results in general topology, Notices of A.M.S. (1984) 25 (1978), A-365.

    Google Scholar 

  14. _____, number of open sets and Boolean algebras with few endomorphisms. Abstracts of A.M.S. 5(1984)

    Google Scholar 

  15. _____, Boolean algebras, General topology and independence results, Abstracts of A.M.S. 5(1984).

    Google Scholar 

  16. _____, Constructions of many complicated uncountable structures and Boolean Algebra, Israel J. Math. 45 (1983), 100–146.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1986 Springer-Verlag

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Shelah, S. (1986). Remarks on the numbers of ideals of Boolean algebra and open sets of a topology. In: Around Classification Theory of Models. Lecture Notes in Mathematics, vol 1182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098509

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  • DOI: https://doi.org/10.1007/BFb0098509

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16448-7

  • Online ISBN: 978-3-540-39788-5

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