On Cauchy condition and related notion of connectedness

Abstract

Connectedness (resp. uniform connectedness [14]) of uniform spaces can be defined in terms of continuous (resp. uniformly continuous) functions to a discrete space requiring that every continuous (resp. uniformly continuous) function to a discrete space has to be constant. Replacing uniformly continuous functions by the strictly weaker notion of Cauchy regular functions [15] we obtain a new notion of connectedness, namely Cauchy connectedness which happens to be an intermediate notion between connectedness and uniform connectedness. We primarily investigate several features of this new notion. Further in metric spaces, we turn our attention to quasi-Cauchy sequences [10] and show that replacing Cauchy regular continuity by ward continuity (functions preserving quasi-Cauchy sequences, see [11]) one again gets back the notion of uniform connectedness. Finally we define WC spaces in line of UC spaces and obtain certain characterizations.