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

Abstract

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 \(w(G)\le \aleph _1\cdot w(X)\), 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.