Closed embeddings of spaces and groups into pseudocompact kR-groups

A well-known theorem of Noble states that each Tychonoff space X is homeomorphic to a closed subspace of a pseudocompact kR-space. We strengthen this result by showing that any Tychonoff space X is homeomorphic to a closed subspace of an abelian pseudocompact kR-group G such that w(G)≤ℵ1·w(X), and if, in addition, X is a precompact group, then X is topologically isomorphic to a closed subgroup of G. It is constructed the first examples of pseudocompact groups G1 and G2 (in fact, they are even countably compact and of weight ℵ2) such that G1 is Ascoli but not a kR-space, and G2 is a kR-space but not a k-space. Under MA+¬CH, we show that any pseudocompact group of weight ℵ1 is Ascoli. These results are proved using topological properties of pseudocompact spaces X of weight ℵ1 and of Σ-products in products of compact spaces. Being motivated by these results and the countably compact part of Noble’s theorem, it is shown by a well-known technique that each countably compact infinite group has a separable countably compact subgroup of cardinality continuum.