Boundedly UC spaces: characterisations and preservation

Abstract

A metric space (X, d) is called a boundedly UC space if every closed and bounded subset of X is a UC space. A metric space (X, d) is called a UC space if each real-valued continuous function on (X, d) is uniformly continuous. In this paper, we study twenty-two equivalent conditions for a metric space to be a boundedly UC space. Also, we study the maps that preserve boundedly UC spaces.

Keywords: Uniform continuity; asymptotic sequence; boundedly UC space; boundedly compact; accumulation point; pseudo-Cauchy; UC-preserving; bounded-covering

Quaestiones Mathematicae 30(2007), 247–262