CLOPEN LINEAR SUBSPACES AND CONNECTEDNESS IN FUNCTION SPACES

Abstract

We aim to study clopen linear subspaces and connectedness properties of the space $C(X)$ of all real-valued continuous functions defined on a metric space $(X,d)$ equipped with various topologies. In particular, we consider the topologies of strong Whitney and strong uniform convergence on bornology. We also examine when these topologies on $C(X)$ are locally convex. While studying clopen subspaces, we give new characterizations for the notion of a shield and for a bornology to be shielded from closed sets.