Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some spaces related to topological inequalities proven by the Erdős-Rado theorem

Author: William G. Fleissner
Journal: Proc. Amer. Math. Soc. 71 (1978), 313-320
MSC: Primary 54A25; Secondary 02K05
MathSciNet review: 0493930
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Erdös-Rado theorem is very useful in proving cardinal inequalities in topology. It has been suggested that certain of these inequalities might be strengthened. We note that trees constructed by Jensen and Gregory using various extra axioms of set theory yield several counterexamples to these suggestions; for example, a space $ X,\vert X\vert = {\omega _2},c(X) = {\omega _1},\chi (X) = \omega $, answering a question of Hajnal and Juhász. We consider the apparently similar relation between $ \vert X\vert,e(X)$, and $ d\Psi (X)$ of Ginsburg and Woods. Using combinatorial consequences of $ V = L$, we construct $ {G_\delta }$ tree families, and establish that, assuming $ V = L$, an infinite cardinal $ \kappa $ is weakly compact $ {\text{iff}}\,d\Psi (X) < \kappa ,{e_a}(X) \subset \kappa {\text{imply}}\vert X\vert < \kappa $.

We consider products of countable chain condition spaces, and show that, using Cohen forcing that ( $ {2^\omega }$ can be anything allowed by König's theorem and there are spaces $ X,Y,c(X) = c(Y) = \omega ,c(X \times Y) = {2^\omega }$). A variation is a space W with the property $ c({W^n}) = {\omega _{n - 1}}$.

References [Enhancements On Off] (What's this?)

  • [De] K. Devlin, Aspects of constructibility, Lecture Notes in Math., vol. 354, Springer-Verlag, Berlin and New York, 1973. MR 0376351 (51:12527)
  • [EH] P. Erdös and A. Hajnal, Unsolved problems in set theory, Proc. Sympos. Pure Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1971, pp. 17-48. MR 0280381 (43:6101)
  • [Ga] F. Galvin, Chain conditions and products, Fund. Math. (to appear). MR 585558 (81m:03058)
  • [GW$ _{1}$] J. Ginsburg and R. G. Woods, On the cellularity of $ \beta X - X$, Proc. Amer. Math. Soc. 57 (1976), 151-154. MR 0407789 (53:11560)
  • [GW$ _{2}$] -, A cardinal inequality for topological spaces involving closed discrete sets, Proc. Amer. Math. Soc. 64 (1977), 357-360. MR 0461407 (57:1392)
  • [Gr] J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), 663-671. MR 0485361 (58:5208)
  • [Je] R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308. MR 0309729 (46:8834)
  • [Ju] I. Juhász, Cardinal functions in topology, Math. Centrum, Amsterdam, 1971.
  • [Kp$ _{1}$] D. Kurepa, Ensembles linéaires et une classe de tableaux ramified, Publ. Math. Univ. Belgrade 6 (1936), 129-160.
  • [Kp$ _{2}$] -, The Cartesian multiplication and cellularity numbers, Publ. Inst. Math. (Beograd) (NS) 2 (1962), 121-139. MR 0177894 (31:2152)
  • [Ro] J. Roitman, Adding a Cohen or random real: Topological consequences and effect on Martin's axiom, Fund. Math. (to appear).
  • [Ru] M. E. Rudin, Souslin's conjecture, Amer. Math. Monthly 76 (1969), 1113-1119. MR 0270322 (42:5212)
  • [Sh] B. Šapirovskiĭ, Canonical sets and character. Density and weight in compact spaces, Soviet Math. Dokl. 15 (1974), 1282-1287.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54A25, 02K05

Retrieve articles in all journals with MSC: 54A25, 02K05

Additional Information

Keywords: Cardinal functions in topology, trees, $ V = L$, Erdös-Rado theorem, cellularity, character, spread, $ {G_\delta }$ diagonal, products of ccc spaces, Cohen forcing
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society