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 k R -space. We strengthen this result by showing that any Tychonoff space X is homeomorphic to a closed subspace of an abelian pseudocompact kR k R -group G such that w(G)≤ ₁· w(X) w ( G ) ≤ ℵ 1 · 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 G 1 and G₂ G 2 (in fact, they are even countably compact and of weight ₂ ℵ 2 ) such that G₁ G 1 is Ascoli but not a kR k R -space, and G₂ G 2 is a kR k R -space but not a k -space. Under MA+ CH MA + ¬ CH , we show that any pseudocompact group of weight ₁ ℵ 1 is Ascoli. These results are proved using topological properties of pseudocompact spaces X of weight ₁ ℵ 1 and of Σ 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.