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

DOI:
https://doi.org/10.1090/S0002-9939-1978-0493930-4

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 $X,|X| = {\omega _2},c(X) = {\omega _1},\chi (X) = \omega$, answering a question of Hajnal and Juhász. We consider the apparently similar relation between $|X|,e(X)$, 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 ${\text {iff}} d\Psi (X) < \kappa ,{e_a}(X) \subset \kappa {\text {imply}}|X| < \kappa$. 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 $c({W^n}) = {\omega _{n - 1}}$.

- Keith J. Devlin,
*Aspects of constructibility*, Lecture Notes in Mathematics, Vol. 354, Springer-Verlag, Berlin-New York, 1973. MR**0376351** - P. Erdős and A. Hajnal,
*Unsolved problems in set theory*, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 17–48. MR**0280381** - Fred Galvin,
*Chain conditions and products*, Fund. Math.**108**(1980), no. 1, 33–48. MR**585558**, DOI https://doi.org/10.4064/fm-108-1-33-48 - John Ginsburg and R. Grant Woods,
*On the cellularity of $\beta X-X$*, Proc. Amer. Math. Soc.**57**(1976), no. 1, 151–154. MR**407789**, DOI https://doi.org/10.1090/S0002-9939-1976-0407789-2 - John Ginsburg and R. Grant Woods,
*A cardinal inequality for topological spaces involving closed discrete sets*, Proc. Amer. Math. Soc.**64**(1977), no. 2, 357–360. MR**461407**, DOI https://doi.org/10.1090/S0002-9939-1977-0461407-7 - John Gregory,
*Higher Souslin trees and the generalized continuum hypothesis*, J. Symbolic Logic**41**(1976), no. 3, 663–671. MR**485361**, DOI https://doi.org/10.2307/2272043 - R. Björn Jensen,
*The fine structure of the constructible hierarchy*, Ann. Math. Logic**4**(1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR**309729**, DOI https://doi.org/10.1016/0003-4843%2872%2990001-0
I. Juhász, - Đuro Kurepa,
*The Cartesian multiplication and the cellularity number*, Publ. Inst. Math. (Beograd) (N.S.)**2(16)**(1963), 121–139 (1963). MR**177894**
J. Roitman, - Mary Ellen Rudin,
*Souslin’s conjecture*, Amer. Math. Monthly**76**(1969), 1113–1119. MR**270322**, DOI https://doi.org/10.2307/2317183
B. Šapirovskiĭ,

*Cardinal functions in topology*, Math. Centrum, Amsterdam, 1971. D. Kurepa,

*Ensembles linéaires et une classe de tableaux ramified*, Publ. Math. Univ. Belgrade

**6**(1936), 129-160.

*Adding a Cohen or random real*:

*Topological consequences and effect on Martin’s axiom*, Fund. Math. (to appear).

*Canonical sets and character. Density and weight in compact spaces*, Soviet Math. Dokl.

**15**(1974), 1282-1287.

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,
<IMG WIDTH="62" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$V = L$">,
Erdös-Rado theorem,
cellularity,
character,
spread,
<!– MATH ${G_\delta }$ –> <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${G_\delta }$"> diagonal,
products of ccc spaces,
Cohen forcing

Article copyright:
© Copyright 1978
American Mathematical Society