Math Graduate Student Seminar
Given two vector spaces, it's not hard to check whether one embeds into the other, since it suffices to compare their dimensions. However, such an embedding may not always be "definable"; for instance, it may require the axiom of choice. In the spirit of descriptive set theory, it is thus natural to consider Borel embeddings of Polish vector spaces, and more generally, Polish modules. We show under the Borel embedding preorder that there is a minimum uncountable-dimensional Polish vector space, and that there is a countable set of uncountable Polish abelian groups such that every uncountable Polish abelian group contains one of these. This is joint work with Joshua Frisch.