Some spaces related to topological inequalities proven by the ErdősRado theorem
Author:
William G. Fleissner
Journal:
Proc. Amer. Math. Soc. 71 (1978), 313320
MSC:
Primary 54A25; Secondary 02K05
MathSciNet review:
0493930
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The ErdösRado 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 , answering a question of Hajnal and Juhász. We consider the apparently similar relation between , and of Ginsburg and Woods. Using combinatorial consequences of , we construct tree families, and establish that, assuming , an infinite cardinal is weakly compact . We consider products of countable chain condition spaces, and show that, using Cohen forcing that ( can be anything allowed by König's theorem and there are spaces ). A variation is a space W with the property .
 [De]
Keith
J. Devlin, Aspects of constructibility, Lecture Notes in
Mathematics, Vol. 354, SpringerVerlag, BerlinNew York, 1973. MR 0376351
(51 #12527)
 [EH]
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
(43 #6101)
 [Ga]
Fred
Galvin, Chain conditions and products, Fund. Math.
108 (1980), no. 1, 33–48. MR 585558
(81m:03058)
 [GW]
John
Ginsburg and R.
Grant Woods, On the cellularity of
𝛽𝑋𝑋, Proc. Amer. Math.
Soc. 57 (1976), no. 1, 151–154. MR 0407789
(53 #11560), http://dx.doi.org/10.1090/S00029939197604077892
 [GW]
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 0461407
(57 #1392), http://dx.doi.org/10.1090/S00029939197704614077
 [Gr]
John
Gregory, Higher Souslin trees and the generalized continuum
hypothesis, J. Symbolic Logic 41 (1976), no. 3,
663–671. MR 0485361
(58 #5208)
 [Je]
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 0309729
(46 #8834)
 [Ju]
I. Juhász, Cardinal functions in topology, Math. Centrum, Amsterdam, 1971.
 [Kp]
D. Kurepa, Ensembles linéaires et une classe de tableaux ramified, Publ. Math. Univ. Belgrade 6 (1936), 129160.
 [Kp]
Đuro
Kurepa, The Cartesian multiplication and the cellularity
number, Publ. Inst. Math. (Beograd) (N.S.) 2 (16)
(1963), 121–139 (1963). 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]
Mary
Ellen 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), 12821287.
 [De]
 K. Devlin, Aspects of constructibility, Lecture Notes in Math., vol. 354, SpringerVerlag, 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. 1748. MR 0280381 (43:6101)
 [Ga]
 F. Galvin, Chain conditions and products, Fund. Math. (to appear). MR 585558 (81m:03058)
 [GW]
 J. Ginsburg and R. G. Woods, On the cellularity of , Proc. Amer. Math. Soc. 57 (1976), 151154. MR 0407789 (53:11560)
 [GW]
 , A cardinal inequality for topological spaces involving closed discrete sets, Proc. Amer. Math. Soc. 64 (1977), 357360. MR 0461407 (57:1392)
 [Gr]
 J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), 663671. MR 0485361 (58:5208)
 [Je]
 R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229308. MR 0309729 (46:8834)
 [Ju]
 I. Juhász, Cardinal functions in topology, Math. Centrum, Amsterdam, 1971.
 [Kp]
 D. Kurepa, Ensembles linéaires et une classe de tableaux ramified, Publ. Math. Univ. Belgrade 6 (1936), 129160.
 [Kp]
 , The Cartesian multiplication and cellularity numbers, Publ. Inst. Math. (Beograd) (NS) 2 (1962), 121139. 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), 11131119. MR 0270322 (42:5212)
 [Sh]
 B. Šapirovskiĭ, Canonical sets and character. Density and weight in compact spaces, Soviet Math. Dokl. 15 (1974), 12821287.
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
DOI:
http://dx.doi.org/10.1090/S00029939197804939304
PII:
S 00029939(1978)04939304
Keywords:
Cardinal functions in topology,
trees,
,
ErdösRado theorem,
cellularity,
character,
spread,
diagonal,
products of ccc spaces,
Cohen forcing
Article copyright:
© Copyright 1978
American Mathematical Society
