| In Chapter 1, we introduce a notion of universality for subgroups of Polish groups that has both algebraic and topological aspects. More precisely, given a class C of subgroups of a topological group G, we say that a subgroup H in C is a universal C subgroup of G if every subgroup K in C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respect to a natural pre-order on the set of subgroups of G. In Chapter 2, we show that for any Polish group G, the countable power of G has a universal analytic subgroup. Moreover, if G is locally compact, then the countable power of G also contains universal sigma-compact and compactly generated subgroups. We prove a weaker version of this in the non-locally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universal analytic, sigma-compact and compactly generated subgroups. As an aside, we explore the relationship between the classes of sigma-compact and compactly generated subgroups and give conditions under which the two coincide.;In Chapter 3, we study universal dense and co-null sets for the classes of G-delta and F-sigma sets, respectively. Specifically, one says that a subset A of X x Y is a universal dense G-delta for Y (resp. co-null F-sigma) if the vertical cross-sections of A are exactly the dense G-delta (resp. co-null F-sigma) subsets of Y. We discuss the relatioship between selection theorems for the product space X x Y and the existence of such universal sets. In the process, we refine a selection theorem of Debs and Saint-Raymond. These results relate to a question of R. D. Mauldin. |
