A comparative survey of selected results and open problems concerning topological groups, fields and vector spaces

It is natural to expect that the behaviour of some topological properties tends to improve in the presence of an additional algebraic structure interacting with the topology (for example, in topological groups, topological fields, or topological vector spaces). The purpose of this survey is to compare topological groups, topological vector spaces and topological fields as to how far each of these classes of spaces is from the class of Tychonoff spaces. In other words, we want to compare the degree of how much of an additional strain an algebraic structure of a group, vector space or field which agrees with the topology of the space imposes on the topology of that space. We cover selected results and open problems related to normality-type properties, covering properties, Cartesian products, homeomorphic embeddings and dimension theory.