Weakly continuously Urysohn spaces
All spaces are assumed to be regular T\(_1\). A space \(X\) is said to be \textit{continuously Urysohn} (cU) if there is a continuous function \(\varphi\) of \(X^2\setminus \Delta\) into the space of bounded real-valued functions with the norm topology such that \(\varphi(x,y)(x)\neq \varphi(x,y)(y)\). The authors call a space \(X\) \textit{weakly continuously Urysohn} (wcU) if there is a continuous function \(\varphi:(X^2\setminus \Delta)\times X \to \mathbb{R}\) such that \(\varphi(x,y,x)\neq\varphi(x,y,y)\) [cf. \textit{Ph. Zenor}, Proc. Am. Math. Soc. 135, No.~1, 305--312 (2007; Zbl 1111.54015)]. The implication cU \(\to\) wcU is obvious, but the converse is in question. The authors study properties of wcU spaces and give some examples. They show that a wcU \(w\Delta\)-space has a base of countable order, and as a corollary that a submetacompact wcU \(w\Delta\)-space is a Moore space, which generalizes Stepanova's metrization result that a paracompact cU p-space is metrizable.
base of countable order, monotonically normal
scientific article; zbMATH DE number 5573808