Algebraic structure of compact-like topological groups and convergence properties

设X是拓扑群G的一个子空间，我们称X拓扑生成群G，只要X代数生成的G的最小子群在G中稠密。在G拓扑生成子集X中，存在一个具有最小可能权w(X)的闭子集，我们称这个权为拓扑生成权。研究了紧群G的拓扑生成权，得到了如下结果：定理1.零维紧交换群G的拓扑生成权与G的拓扑生成权重合.定理2连通紧交换群G的拓扑生成权与G的权的欧米根重合(这里k的基数k的欧米根是S使得S的欧米伽幂超过k的最小可能基数)定理3.紧交换群G的拓扑生成权等于连通分支c(G)的拓扑生成权的乘积)和G/c(G)的权。我们还研究了可数紧Abel群的代数结构。特别地，我们研究了大小至多为2^c的Abel群G是否允许可数紧群拓扑。(这里c表示连续统的基数。)利用强迫的方法，我们构造了集合论的Zermelo-Fraenkel公理模型M，其中定理4成立。定理4对于Abel群G，下列条件是等价的：(I)G是可分可数紧群拓扑，(Ii)G是可分可数紧群拓扑，(Iii)G是没有无限紧子集的可遗传可分可数紧群拓扑，(Iv)G的大小至多为2^c，且满足条件Ps和Cc。定理5对于无限Abelian群，下列条件是等价的：(I)G有可分伪紧群拓扑(Ii)G的基数介于c和2^c之间，且满足条件Ps。

Let X be a subspace of a topological group G. We say that X topologically generates G provided that the smallest subgroup of G algebraically generated by X is dense in G. Among all closed subsets X of G topologically generating G there exists one that has the smallest possible weight w(X), and we call this weight topologically generating weight. We investigated the topologically generating weight of a compact group G and obtained the following results :Theorem 1.Topologically generating weight of a zero-dimensional compact Abelian group G coincides with the weight of G.Theorem 2.Topologically generating weight of a connected compact Abelian group G coincides with the omega-root of weight of G. (Here the omega-root of a cardinal k is the smallest possible cardinal s such that the omega power of s exceeds k.)Theorem 3.Topologically generating weight of a compact Abelian group G is equal to the product of the topologically generating weight of the connected component c(G) of G and the weight of G/c(G).We also study algebraic structure of countably compact Abelian group. In particular, we investigate whether an Abelian group G of size at most 2^c admits a countably compact group topology. (Here c denotes the cardinality of the continuum.) Using forcing, we have constructed a model M of Zermelo-Fraenkel Axioms of Set Theory in which the following Theorm 4 holds.Theorem 4.For an Abelian group G the following conditions are equivalent :(i)G admits a separable countably compact group topology,(ii)G admits a hereditarily separable countably compact group topology,(iii)G admits a hereditarily separable countably compact group topology without infinite compact subsets,(iv)G has size at most 2^c and satisfies conditions Ps and CC.Theorem 5.For an infinite Abelian group the following conditions are equivalent :(i)G has a separable pseudocompact group topology,(ii)G has cardinality between c and 2^c and satisfies condition Ps.