[EN] Based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. Compactness here is sometimes viewed splitting into “k-Lindelöfness” and ”k-compactness” for cardinals k. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result of Som Naimpally, to who this article is hearty dedicated. The research in this paper was partially conducted while the second author was visiting Professor T. Nogura at Ehime University with the financial support of the Ministry of Education of Japan. Castro Pereira, I.; Tomita, A. (2004). A countably compact free Abelian group whose size has countable cofinality. Applied General Topology. 5(1):97-101. doi:10.4995/agt.2004.1998. Van Douwen, E. K. (1980). The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality. Proceedings of the American Mathematical Society, 80(4), 678-678. doi:10.1090/s0002-9939-1980-0587954-5 P. B. Koszmider, A. H. Tomita and S. Watson, Forcing countably compact group topologies on a larger free Abelian group, Topology Proc. 25 (Summer 2000), 563–574. K. Kunen, Set theory. An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, 102. North-Holland Publishing Co., Amsterdam, 1980. xvi+313. M. G. Tkachenko, Countably compact and pseudocompact topologies on free abelian groups, Izvestia VUZ. Matematika 34 (1990), 68–75.

A countably compact free Abelian group whose size has countable cofinality

Abstract