Inductive ring topologies

Abstract

Since topological algebra is the study of algebraic structures with topologies for which the operations are continuous, a natural question for the topological algebraist to ask is whether a given structure admits any such topologies whatever, other than the discrete and indiscrete ones. The question has been answered for some classes of structures. For example, Kertesz and Szele [7] prove that every infinite abelian group admits a nondiscrete, Hausdorff group On the other hand, Hanson [5] gives an example of an infinite groupoid which admits only the two trivial topologies mentioned above. Our purpose here is to answer this question for infinite fields, proving that every infinite field admits a nondiscrete, Hausdorff field This will be done by affirmatively answering the question for two classes of commutative rings: the first being all integral domains with a certain cardinality condition (§3), and the second, all rings which are the union of a chain of subrings with certain properties (§4). These two classes will be shown to include all infinite fields (§5). Our method of proof will make use of an procedure first used by Hinrichs [6] to prove the existence of certain unusual topologies on the integers. The procedure is described in §1, where we define what we mean by an inductive ring topology. In §§7 and 8, we turn our attention to some further applications of topologies, showing first how they can be used to construct interesting examples of topologies on the integers and rational numbers. We use them to get proofs that there are uncountably many, and non-first countable ring topologies on all the rings considered in §3 and §4. We also show how characterizations can be obtained for several classes of topologies on fields using modifications of the method. A supplement to our discussion of field topologies comes in §6, where we characterize those fields which admit nondiscrete, Hausdorff, locally bounded topologies. The methods used here, however, are those of valuation theory. When we say that a topology S~ is a ring topology on a ring A, we mean that the mappings (a, b)^-a—b and (a, b)-*ab from Ax A into A are continuous. &~ is a field topology on a field K if it is a ring topology, and in addition, the mapping a -*■ a'1 is continuous on K~{0}.