A solution to a problem of E. Michael
A product space \(X\times Y\) is defined to be rectangularly normal if every continuous real-valued function on a closed rectangle \(A\times B\) in \(X\times Y\) can be continuously extended over \(X\times Y\). It is essentially proved by \textit{K. Morita} [Fund. Math. 88, 1-6 (1975; Zbl 0304.55009)], \textit{M. Starbird} [Stud. Topol. 559-564 (1975; Zbl 0309.54014)] and \textit{L. E. Mednikov} [Dokl. Akad. Nauk SSSR 222, 1287- 1290 (1975; Zbl 0334.54005)] that the product \(X\times Y\) of a normal space X with a locally compact metric space Y is rectangularly normal. In connection with the converse of this result the purpose of this paper is to find a normal space X such that the product \(X\times M\) with a metric space M is rectangularly normal iff M is locally compact, and consequently \(X\times Q\), where \(Q=the\) space of rationals, is not rectangularly normal; the latter answers a question raised by E. Michael. Indeed, as such a space, the author mentions a normal non-countably functionally Katětov space that was already given by the author and \textit{M. L. Wage} [Fundam. Math. 109, 175-187 (1980; Zbl 0443.54015)], by proving a theorem that \(X\times M\) is rectangularly normal for some non- locally compact metric space M iff X is countably functionally Katětov. Finally, an example of some non-rectangular product in the sense of \textit{B. A. Pasynkov} [Dokl. Akad. Nauk SSSR 221, 291-294 (1975; Zbl 0334.54024)] is constructed.
normal non-countably functionally Katětov space
scientific article; zbMATH DE number 3890317