On the uniform paracompactness of the product of two uniform spaces (Q789767)

The concept of uniform paracompactness was introduced by \textit{M. D. Rice} [Proc. Am. Math. Soc. 62, 359-362 (1977; Zbl 0353.54011)]. The main result concerns the behavior of this property under products. The author generalizes a theorem of A. Hohti, stated for metric spaces, to uniform spaces: The product \({\mathcal U}\times {\mathcal V}\) of two uniformly paracompact uniform spaces \({\mathcal U}\), \({\mathcal V}\), which are both first- countable as topological spaces, is uniformly paracompact, if and only if either both spaces are locally compact or at least one of them is compact. Some interesting examples are given.