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.