Forcing hereditarily separable compact-like group topologies on Abelian groups

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• 作者：Dikranjan ; Dikran ; Shakhmatov ; Dmitri
• 关键词：Topological group ; Countably compact ; Pseudocompact ; Hereditarily separable ; Convergent sequence ; S-space ; Consistency results ; Independence results ; Forcing
• 刊名：Topology and its Applications
• 出版年：2005
• 出版时间：June 1, 2005
• 年：2005
• 卷：151
• 期：1-3
• 页码：2-54
• 全文大小：431 K

文摘

Let 70234ca1944e4""> denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most ) which admit:

(i) a hereditarily separable group topology,

(ii) a group topology making G into an S-space,

(iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets),

(iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary).

As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired).

We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.