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Almost disjoint families and property (a)

Paul Szeptycki; Jerry Vaughan

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 3, page 229-240
  • ISSN: 0016-2736

Abstract

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We consider the question: when does a Ψ-space satisfy property (a)? We show that if | A | < p then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality p which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).

How to cite

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Szeptycki, Paul, and Vaughan, Jerry. "Almost disjoint families and property (a)." Fundamenta Mathematicae 158.3 (1998): 229-240. <http://eudml.org/doc/212313>.

@article{Szeptycki1998,
abstract = {We consider the question: when does a Ψ-space satisfy property (a)? We show that if $|A| < p$ then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality $p$ which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).},
author = {Szeptycki, Paul, Vaughan, Jerry},
journal = {Fundamenta Mathematicae},
keywords = {property (a), density; extent; almost disjoint families; Ψ-space; CH; GCH; Martin's Axiom; $p = c$; Cohen forcing; Q-set; weakly inaccessible cardinal.; property (a); density; almost disjoint family; -space; Martin's axiom},
language = {eng},
number = {3},
pages = {229-240},
title = {Almost disjoint families and property (a)},
url = {http://eudml.org/doc/212313},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Szeptycki, Paul
AU - Vaughan, Jerry
TI - Almost disjoint families and property (a)
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 3
SP - 229
EP - 240
AB - We consider the question: when does a Ψ-space satisfy property (a)? We show that if $|A| < p$ then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality $p$ which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).
LA - eng
KW - property (a), density; extent; almost disjoint families; Ψ-space; CH; GCH; Martin's Axiom; $p = c$; Cohen forcing; Q-set; weakly inaccessible cardinal.; property (a); density; almost disjoint family; -space; Martin's axiom
UR - http://eudml.org/doc/212313
ER -

References

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  1. [1] M. G. Bell, On the combinatorial principal P(c), Fund. Math. 114 (1981), 149-157. Zbl0581.03038
  2. [2] E. K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 111-167. 
  3. [3] R. Engelking, General Topology, PWN, Warszawa, 1977. 
  4. [4] W. G. Fleissner and A. W. Miller, On Q-sets, Proc. Amer. Math. Soc. 78 (1980), 280-284. 
  5. [5] D. H. Fremlin, Consequences of Martin's Axiom, Cambridge Univ. Press, Cambridge, 1984. Zbl0551.03033
  6. [6] L. Gillman and M. Jerison, Rings of Continuous Functions, van Nostrand, Princeton, 1960. Zbl0093.30001
  7. [7] S. H. Hechler, Short complete nested sequences in βNNand small maximal almost-disjoint families, Gen. Topology Appl. 2 (1972), 139-149. Zbl0246.02047
  8. [8] R. E. Hodel, Cardinal Functions I, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 1-61. 
  9. [9] W. Just, M. V. Matveev and P. J. Szeptycki, Some results on property (a), Topology Appl., to appear. Zbl0944.54014
  10. [10] K. Kunen, Set Theory, North-Holland, 1980. 
  11. [11] M. V. Matveev, Absolutely countably compact spaces, Topology Appl. 58 (1994), 81-92. Zbl0801.54021
  12. [12] M. V. Matveev, On feebly compact spaces with property (a), preprint. 
  13. [13] M. V. Matveev, Some questions on property (a), Questions Answers Gen. Topology 15 (1997), 103-111. Zbl1002.54016
  14. [14] M. E. Rudin, I. Stares and J. E. Vaughan, From countable compactness to absolute countable compactness, Proc. Amer. Math. Soc. 125 (1997), 927-934. Zbl0984.54027

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