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Products, countable compactness and p-limits
by
Artur H. Tomita
Universidade de Sao Paulo
Nov\' ak [Fund. Math., 1953] and Terasaka [Osaka Math. J, 1952] showed that there exists a countably compact space whose product is not countably compact. Many interesting questions related to this have been investigated since.
The concepts of p-limit and p-compactness, due to Berstein [Fund. Math, 1970], are very useful in the study of compactness.
Definition 1. Given a free ultrafilter p, a sequence {xn: n in \omega} in X has x as a p-limit if {n in \omega: xn in U} in p for every neighborhood of x in X.
Definition 2. A space X is p-compact if every sequence in X has a p-limit in X.
Basic properties include:
- every accumulation point of a sequence is a p-limit for some p - p-compactness, for a fixed p, is a productive property and - p-compact spaces are countably compact.
Let us mention three kinds of problems about countable compactness and products that have been studied in the literature:
A) Large infinite families of spaces in which the full product is not countably compact but the product of subfamilies of smaller cardinality are not.
Many questions of this sort have been asked by Comfort in the seventies. Some papers in this topic include: Scarborough and Stone [TAMS, 1966], Ginsburg and Saks [Pacific J. Math, 1975], Saks [TAMS, 1978] and Yang [Top. Proc., 1985].
B) Finite families of topological groups in which the full product is not countably compact but the product of subfamilies of smaller cardinality are not.
The known examples deal with finite powers and finite products using some form of Martin's Axiom: E. van Douwen [TAMS, 1980], Hart and van Mill [TAMS, 1991], Tomita [CMUC, 1996] and Tomita [Top. Appl., 1999]. A manuscript of Tomita and Watson shows that, under MAcountable, there exists, for each integer n > 1, a countably compact group G without non-trivial convergent sequences such that Gn is countably compact but Gn+1 is not.
C) Preservation of countable compactness of p-compact spaces, for different p's.
The existence of a p-compact space and a q-compact space whose product is not countably compact is independent of ZFC (Saks [Top. Proc., 1979] for the existence and Garcia-Ferreira [Top. Appl., 1993] for the non-existence).
Garcia-Ferreira (question 482 in the Open Problems in Topology) asked for an example which would generalize van Douwen's in B) and Saks' example in C).More precisely, he asked if Martin's Axiom imply the existence of a p-compact group and a q-compact group whose product is not countably compact. This question has been answered by Tomita and Watson in the positive under MAcountable.
In this talk , we discuss some examples, relating: A) and B; A) and C); and B) and C) for countable (finite or infinite) products.
Date received: June 29, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-55.