Algebraic structure of compact-like topological groups and convergence properties

设X是拓扑群G的一个子空间。我们说，如果由X代数生成的G的最小子群在G中是稠密的，则X拓扑生成G。在拓扑生成G的G的所有闭子集X中，存在一个权值w(X)最小的闭子集，我们称这个权值为拓扑生成权值。我们研究了紧群G的拓扑生成权，得到了以下结果：定理1。拓扑生成零维紧阿贝尔群G的权值与G的权值重合。在拓扑上生成连通紧阿贝尔群G的权值与G的权值的根一致（这里k的根是使s的幂超过k的最小可能的基数s）。定理3。紧阿贝尔群G的拓扑生成权等于G的连通分量c(G)的拓扑生成权与G/c(G)的权的乘积。研究了可数紧阿贝尔群的代数结构。特别地，我们研究了一个最大为2^c的阿贝尔群G是否存在可数紧群拓扑。（这里c表示连续体的基数。）利用强迫，我们构造了集合论Zermelo-Fraenkel公理的一个模型M，其中定理4成立。定理4。对于一个阿别群G，下列条件是等价的：(i)G允许一个可分可数紧群拓扑，（ii）G允许一个遗传可分可数紧群拓扑，（iii）G允许一个没有无限紧子集的遗传可分可数紧群拓扑，（iv）G的大小不超过2^c，并且满足条件p和cc。对于无限阿贝尔群，下列条件是等价的：(i)G具有可分离的伪紧群拓扑，（ii）G具有介于c和2^c之间的基数，并且满足条件p。

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.