Metric characterization of connectedness for topological spaces

Ittay Weiss

Published 2014-11-24Version 1

Connectedness, path connectedness, and uniform connectedness are well-known concepts. Conceptually speaking there is a substantial difference between connectedness and the other two notions, namely connectedness is defined as the absence of disconnectedness, while path connectedness and uniform connectedness are defined in terms of connecting paths and connecting walks, respectively. In compact metric spaces uniform connectedness and connectedness are well-known to coincide, thus the apparent conceptual difference between the two notions disappears. We present a two-fold generalization of this result by eliminating compactness and considering Flagg's continuity spaces. The resulting connectedness criterion is shown to be valid for all topological spaces, thus completely eradicating the conceptual difference between connectedness and path connectedness.