Let F F be a complete topological field. We undertake a study of the ring C ( X , F ) C(X,F) of all continuous F F -valued functions on a topological space X X whose topology is determined by C ( X , F ) C(X,F) , in that it is the weakest making each function in C ( X , F ) C(X,F) continuous, and of the ring C ∗ ( X , F ) {C^\ast }(X,F) of all continuous F F -valued functions with relatively compact range, where the topology of X X is similarly determined by C ∗ ( X , F ) {C^\ast }(X,F) . The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of X X in the former case and of the Stone-Čech compactification of X X in the latter. Most attention is given to the case where F F and X X are ultraregular; in this case we determine conditions on F F that permit a development parallel to the classical theory where F F is the real number field. One example of such conditions is that the cardinality of F F be nonmeasurable and that the topology of F F be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirota’s theorem concerning the bornologicity of C ( X ) C(X) is obtained.
We assume familiarity with the definitions and basic properties of linearly compact rings and modules as contained in [6] or [3, Exercises 14 20,. In § 1 we show that every linearly compact topology on a module is weaker than a unique, maximal linearly compact topology. In § 2 we apply the results of § 1 to a discussion of the circumstances under which the following statements about a linearly compact topological module E over a linearly compact ring A with radical r are equivalent:(1) the topology of E is stronger than the r-adic topology; (2) rZE is open; (3) rE is open and finitely generated; (4) r--E is open and finitely generated. Our results, applied to linearly compact rings, yield improvements of known theorems, even in the compact case. In §3 we give conditions under which a module, linearly compact for the discrete topology, is necessarily noetherian. In particular, we show that if 0 r"= (0) where r is the radical of a (not necessarily commutative) n>l ring A with identity, then A is linearly compact for the discrete topology if and only if A is (left) noetherian, A/r is (left) artinian, and A is complete for the r-adic topology. I. Maximal Linearly Compact TopologiesIf Y is a linearly compact topology on an A-module E, of all the Hausdorff linear topologies on E weaker than :-there is a weakest, which we shall denote by ,Y, [6, Theorem 2], [3, Exercises 17-18, p. 110] ; E, equipped with ~-,, is linearly compact, of course. Here we shall show that of all the linearly compact topologies on E stronger than Y, there is a strongest. We recall first that if 5: and Y are linearly compact topologies on E and if 5: is stronger than ~-, then E; 5 e and E; Y have the same closed submodules. Indeed, any subset closed for Y is clearly closed for 09~; conversely, since the identity mapping from E;5 e onto E; .Y" is continuous, a submodule closed for~ is linearly compact and hence closed for 5-. Theorem 1. Let ~--be a linearly compact topology on an A-module E.Of all the linearly compact topologies on E stronger than J there is a
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.