Metric spaces on which continuous functions are ``almost'' uniformly continuous
In this paper, the author discusses the relationship among the notions of continuous functions, almost-bounded functions, almost-uniformly continuous functions and Cauchy-continuous functions, and proves that the image of a Cauchy sequence in a metric space under a continuous and almost-uniformly continuous function has a Cauchy subsequence, and the image of a totally bounded metric space under a continuous and almost-uniformly continuous function is totally bounded. Some characterizations of Lebesgue metric spaces are given. The author also considers some properties of \(AUC\) metric spaces. The following results are proved: (1) Every \(AUC\) metric space is complete; (2) A metric space is compact if and only if it is totally bounded and \(AUC\); (3) Let \(\mathbf{X}=(X,d)\) be a metric space. The following are equivalent: (i) \(\mathbf{X}\) is \(AUC\); (ii) \(\mathbf{X}\) is weakly Lebesgue; (iii) \(\mathbf{X}\) is weakly countably Lebesgue; (iv) For every metric space \(\mathbf{Y}=(Y,\rho)\), every continuous function \(f:\mathbf{X}\rightarrow\mathbf{Y}\) is almost-uniformly continuous.
scientific article; zbMATH DE number 6804710
scientific article