Abstract
We determine those regular cardinals κ with the property that for each increasing κ-chain of first countable spaces there is a compatible first countable topology on the union of the chain. AssumingV=L any such κ must be weakly compact. It is relatively consistent with a supercompact cardinal that each κ>w 1 has the property. The proofs exploit the connection with interesting families of integer-valued functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. P. Burgess,Forcing, inHandbook of Mathematical Logic (J. Barwise ed.), North-Holland, Amsterdam, 1977.
K. J. Devlin,The Axiom of Constructibility, Springer-Verlag, Heidelberg, 1984.
I. Juhász,Cardinal Functions in Topology — 10 Years Later, Mathematical Centre, Amsterdam, 1981.
I. Juhász and Z. Szentmiklóssy,Increasing strengthenings of cardinal function inequalities, Fund. Math., to appear.
K. Kunen,Set Theory, An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980.
M. Magidor,On the role of supercompact and extendible cardinals in logic, Isr. J. Math.10 (1971), 147–171.
W. J. Mitchell,Aronszajn trees and the independence of the transfer property, Ann. Math. Logic5 (197), 21–46.
S. Todorčević,Families of parallel functions, unpublished manuscript, 1986.
Author information
Authors and Affiliations
Additional information
Research of the second author supported by OTKA grant no. 1805. Research of the remaining authors partially supported by NSERC of Canada.
Rights and permissions
About this article
Cite this article
Dow, A., Juhász, I. & Weiss, W. Integer-valued functions and increasing unions of first countable spaces. Israel J. Math. 67, 181–192 (1989). https://doi.org/10.1007/BF02937294
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02937294