Epi-reflective properties of the Bohr compactification (Q2764172)

For a given locally compact Abelian group (LCA group for short) \(G\), denote by \(G^+\) the group \(G\) endowed with its Bohr topology, the one induced by the set of all continuous characters of \(G\). Answering a question of \textit{E. K. van Douwen} [Topology Appl. 34, 69-91 (1990; Zbl 0696.22003)], the authors characterize in terms of \(G\) when the group \(G^+\) belongs to the epireflective hull of the real numbers or, in other words, when \(G^+\) is realcompact. This characterization proves among other things that \(G^+\) is realcompact if and only if \(G\) is realcompact and that this happens exactly if \(\kappa(G)\) (the minimal cardinality of a cover of \(G\) by compact sets) is not Ulam-measurable. NEWLINENEWLINENEWLINEAs a consequence of this characterization and of further results on the extendability of continuous functions from subspaces of \(G^+\), the authors prove that the Bohr topology of an LCA group \(G\) is hereditarily realcompact exactly if \(G\) is metrizable and \(|G|\leq {\mathfrak c}\). This shows that some LCA groups (discrete groups of cardinality \({\mathfrak c}^+\) for instance) have realcompact subspaces that are not realcompact in the topology they inherit from \(G^+\).NEWLINENEWLINEFor the entire collection see [Zbl 0974.00041].