Atsuji completions: Equivalent characterisations

A metric space ( X , d ) is called an Atsuji space if every real-valued continuous function on ( X , d ) is uniformly continuous. It is well known that an Atsuji space must be complete. A metric space ( X , d ) is said to have an Atsuji completion if its completion ( X ˆ , d ) is an Atsuji space. In this paper, we study twenty-nine equivalent characterisations for a metric space to have an Atsuji completion.