Products of straight spaces with compact spaces

Abstract

[EN] A metric space X is called straight if any continuous real-valued function which is uniformly continuous on each set of a finite cover of X by closed sets, is itself uniformly continuous. Let C be the convergent sequence {1/n : n ϵ N} with its limit 0 in the real line with the usual metric. In this paper, we show that for a straight space X, X × C is straight if and only if X × K is straight for any compact metric space K. Furthermore, we show that for a straight space X, if X × C is straight, then X is precompact. Note that the notion of straightness depends on the metric on X. Indeed, since the real line R with the usual metric is not precompact, R×C is not straight. On the other hand, we show that the product space of an open interval and C is straight.