| Topology Atlas Document # zaab-04 | Topology Atlas Invited Contributions vol. 1, no. 3 (1996) pp. 21-22. |
University of North Carolina at Greensboro,
Greensboro, NC 27412, USA
http://www.uncg.edu/~vaughanj/
A small cardinal is a cardinal number that is defined as the cardinality of a set that is associated in some way with the set of natural numbers N. The cardinality of N is obviously a small cardinal. Another small cardinal is the cardinality of the set of real numbers, denoted by c (an easy way to associate c with N is to note that c is also the cardinality of the set of all functions from N into N). Small cardinals are studied in set theory, of course, and they are also of much interest in parts of topology and analysis. They can provide the best understanding of certain situations, they can allow some results to be stated in ZFC which otherwise would require more technical statements, and sometimes they arise unexpectedly. We will illustrate these features of small cardinals with the following simply stated questions.
Question 1: What is the smallest number of compact subsets of the irrational numbers needed to cover the irrational numbers?
Question 2: Let X be a Lindelöf space. Is the product of X with the irrational numbers normal?
Concerning Question 1, it is clear that one can cover the irrationals with c compact (singleton) sets. On the other hand, by the Baire category theorem, it is not possible to cover the irrational numbers with countably many compact subsets of irrational numbers. Thus if k denotes the least number of compact sets needed to cover the irrational numbers, it appears that all one can say in ZFC towards answering this question is that k is some uncountable cardinal not larger than c. There is a more precise answer using a small cardinal. In this case it is the small cardinal called d, which was introduced by M. Katetov in 1960. The cardinal d is defined to be the smallest cardinality of a set D of functions from N into N with the property that for every function g:N-->N there exists d in D such that g(n) <= d(n) for all but finitely many n in N.
Theorem 1: The irrational numbers can be covered by d compact subsets of irrationals, and not by fewer than d.
An exact answer to Question 2 is not known, but an old result of E. Michael (interpreted in the setting of small cardinals) gives an answer using the small cardinal b, where b is the smallest cardinality of a set B of function from N to N such that for every function g:N-->N, there exists b in B such that g(n) <= b(n) for infinitely many n in N.
Theorem 2. If b equals the first uncountable cardinal, then there exists a Lindelöf space whose product with the irrationals is not normal.
The cardinal b was essentially introduced by F. Rothberger in 1939 (he studied a property which is tantamount to b). As far as I know, Rothberger was the first person to consider a small cardinal property (other that |N| and c). The best reference for small cardinals in topology is the article The integers and topology by Eric van Douwen in Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan, Eds., North-Holland, Amsterdam-New York-Oxford, 1984. For topology and analysis, see the book by David Fremlin, Consequences of Martin's Axiom, Cambridge University Press, London, 1984. I wrote a brief up-date Small cardinals in topology, in the book Open Problems in Topology, J. van Mill and G. M. Reed, eds., North-Holland, Amsterdam, 1990. From time to time, van Mill and Reed publish in Topology and its Applications information about the current status of the problems in their book.
Received by the editors: December 19, 1995. Production Editor: R. Flagg. Revised October 15, 1997.