It is well-known that a finite set is universal, that is, each Lebesgue measurable set with positive measure contains an affine copy of a finite set. The Erdős similarity conjecture, which remains open, states that there is no infinite universal set. In 2022, Gallagher, Lai, and Weber considered a topological version of this conjecture, defining a set to be topologically universal if each dense G-delta set contains an affine copy of the set. They conjectured that there are no such uncountable sets. In this thesis, we give a full classification of topologically universal sets as a special subfamily of measure zero sets. As a corollary, we prove that the topological Erdős similarity conjecture is independent of ZFC. We generalize this result to arbitrary locally compact Polish groups, and use the measure-category duality to pose and investigate the full-measure Erdős similarity conjecture.