A countably compact free Abelian group whose size has countable cofinality

I. Castro Pereira

Brazil

Universidade de Sao Paulo

Departamento de Matemática

A.H. Tomita

Brazil

Universidade de Sao Paulo

Departamento de Matemática
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Accepted: 2013-12-02

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DOI: https://doi.org/10.4995/agt.2004.1998
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Keywords:

Forcing, Countably compact group, Convergence, Continuum Hypothesis, Countable cofinality, Size

Supporting agencies:

This research was not funded

Abstract:

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.

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References:

W. W. Comfort, K. H. Hofmann and D. Remus, Topological groups and semigroups, Recent progress in general topology (Prague, 1991), 57–144, North-Holland, Amsterdam, 1992.

E. K. van Douwen, The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality, Proc. Amer. Math. Soc. 80 (1980), 678–682. http://dx.doi.org/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. H. Tomita, Two countably compact groups: one of sizeNω and the other of weight Nω Without non-trivial convergent sequences, to appear in Proc. Amer. Math. Soc.

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