Algebraic structure of compact-like topological groups and convergence properties

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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.