Characterization of Cauchy regular functions

Abstract

The underlying theme of this article is the class of all Cauchy regular functions. We primarily emphasize specific preserving properties of Cauchy regular functions. From literature and our recent work [5], we can conclude that every Cauchy regular function preserves total boundedness and Cauchy connectedness. We investigate certain interesting conditions under which every total boundedness- and Cauchy connectedness-preserving function is Cauchy regular. In this paper, an interesting weakening of the notion of completeness, namely pre-straightness is considered. We characterize pre-straight spaces in terms of a suitable metric. Also, we introduce Cauchy separated fibers (CSF), which yield Cauchy regularity of a continuous function on a pre-straight space. Finally, a generalization of Cauchy regular function, namely Cauchy approachable function (CA) is defined along the lines of uniformly approachable functions (UA) [7] and the relation between CA and CSF functions is obtained. Furthermore, we give a possible answer to the open question [Question 3.12], posed in [2].