Bohr topologies and compact function spaces (Q1862082)

Let \(\mathcal{U}\) be a structure for a countable language, and denote by \(\mathcal{U}^\sharp\) the structure \(\mathcal{U}\) with its Bohr topology. A compact structure \(\mathcal{X}\) is nice iff there is a single compact second countable structure \(\mathcal{R}\) such that the set of continuous homomorphisms from \(\mathcal{X}\) into \(\mathcal{R}\) separates the points of \(\mathcal{X}\). A structure \(\mathcal{U}\) (without a topology) is nice iff its Bohr compactification is nice. The authors show that a compact Hausdorff space \(Y\) is Eberlein compact iff \(Y\) is homeomorphic to some closed subspace of \(\mathcal{U}^\sharp\) for some nice \(\mathcal{U}\), and that \(Y\) is Talagrand compact iff \(Y\) is homeomorphic to a subspace of some \(\mathcal{U}^\sharp\).