RT journal article
T1 Regularity in Topological Modules
A1 García Pacheco, Francisco Javier
A2 Matemáticas
K1 topological module
K1 topological ring
K1 normed module
K1 absolutely valued ring
K1 regular closed set
K1 regular open set
K1 closed-unit neighborhood of zero
AB The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory calledstrong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, ifMis a topologicalR-module and m* is an element of M* is a continuous linear functional on M which is open as a map between topological spaces, then m*-1(int(B)) is regular open and m*-1(B) is regular closed, for B any closed-unit zero-neighborhood in R.
PB MDPI
SN 2227-7390
YR 2020
FD 2020-09
LK http://hdl.handle.net/10498/23994
UL http://hdl.handle.net/10498/23994
LA eng
DS Repositorio Institucional de la Universidad de Cádiz
RD 13-nov-2026