Saak Gabriyelyan

Closed embeddings of spaces and groups into pseudocompact k_\mathbb{R}-groups.

‎‎‎A well-known theorem of Noble states that each Tychonoff space X is homeomorphic to a closed subspace of a pseudocompact k_\mathbb{R}-space. We strengthen this result by showing that any Tychonoff space X is homeomorphic to a closed subspace of an abelian pseudocompact k_\mathbb{R}-group G such that w(G)\leq \aleph_1\cdot 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 G_1 and G_2 (in fact, they are even countably compact and of weight \aleph_2) such that G_1 is Ascoli but not a k_\mathbb{R}-space, and G_2 is a k_\mathbb{R}-space but not a k-space. Under \mathrm{MA+\lnot CH}, we show that any pseudocompact group of weight \aleph_1 is Ascoli. These results are proved using topological properties of pseudocompact spaces X of weight \aleph_1 and of \Sigma-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.