Left K-Cauchy regular maps between quasi-metric spaces

This article presents a systematic study of a class of maps between quasi-metric spaces that preserve left K-Cauchy sequences. We call such maps left K-Cauchy regular maps. Several characterizations of these maps have been given in terms of the structural properties of the underlying quasi-metric spaces. In particular, we characterize left K-Cauchy regular maps in terms of hereditarily precompact sets by using a suitable version of the classical Efremovic Lemma in the setting of quasi-metric spaces. In addition, we examine the relationship of left K-Cauchy regular maps with (forward) uniformly continuous and continuous maps and prove extension theorems for such maps.