On weak normality of space \(X^ 2\smallsetminus\Delta\) (Q1358552)

\textit{A. V. Arkhangel'skij} and the author proved [Topology Appl. 35, 121-126 (1990; Zbl 0707.54018)] that a compact space \(X\) is first-countable provided that \(X^2\smallsetminus\Delta\) is normal. In this paper the author shows that normality can be replaced by a weaker separation property. Namely, one can merely assume that any two disjoint closed subsets of \(X^2\smallsetminus\Delta\) can be separated, i.e., mapped to disjoint sets by a continuous function of \(X^2\smallsetminus\Delta\) into a metrizable space.