The Hewitt–Nachbin Completion in Topological Algebra. Some Effects of Homogeneity

Abstract

A space X is called Moscow if the closure of any open set is the union of some family of Gδ-subsets of X. It is established that if a topological ring K of non-measurable cardinality is a Moscow space, then the operations in K can be continuously extended to the Hewitt–Nachbin completion υK of K turning υK into a topological ring as well. A similar fact is established for linear topological spaces. If F is a topological field such that the cardinality of F is non-measurable and the space F is Moscow, then the space F is submetrizable and the space F is hereditarily Hewitt–Nachbin complete. In particular, υF=F. We also show the effect of homogeneity of the Hewitt–Nachbin completion on the commutativity of the Hewitt–Nachbin completion with the product operation.