Uniform density and m -density for subrings of C(X

Abstract

Let C(X) denote the continuous real-valued functions on a topological space X . The question of whether a u -dense subring of C(X) is m -dense is studied in this note. Recall that neighborhoods of a function f in the u -topology are determined by an interval (f−ε,f+ε) for ε a positive number and in the m -topology by intervals (f−e,f+e) for u a positive unit in C(X) . J. Kurzweil [Studia Math. 14 (1954), 214–231, had shown that u -denseness and m -denseness are equivalent for subrings of C(X) closed under bounded inversion. Here, the authors prove that this result is not valid for arbitrary subrings of C(X) . In particular, they show that the property of every u -dense subring being m -dense is equivalent to X being pseudocompact