H-2 - Topological Rings, Division Rings, Fields and Lattices

Abstract

A topological division ring or a topological skew field is a division ring equipped with some division ring topology. A division ring topology on a field R is called a field topology on R, and a field equipped with some field topology is called a topological field. A topological ring is a particular type of an Abelian topological group, and thus all results about (Abelian) topological groups also hold in topological rings. Conversely, every Abelian topological group G can be turned into a topological ring by defining the trivial multiplication on G by g.h = 0 for g, h ∈G. If a topological ring is T0-space (in particular, if a topological ring is Hausdorff), then it is automatically a Tychonoff space. There exists a topological field whose topology is Tychonoff but is not normal, that is, a normal topological field that is not hereditarily normal and a hereditarily normal topological field that is not perfectly normal. A non-trivial ring topology on a division ring is automatically Hausdorff. Moreover, a non-trivial ring topology on a division ring is either connected or totally disconnected.