Fragmentability of groups and metric-valued function spaces

Let (X, tau) be a topological space and let rho be a metric defined on X. We shall say that (X, tau) is fragmented by p if whenever epsilon > 0 and A is a nonempty subset of X there is a tau-open set U such that U boolean AND A not equal empty set and rho - diam(U boolean AND A) < epsilon. In this paper we consider the notion of fragmentability, and its generalisation sigma-fragmentability, in the setting of topological groups and metric-valued function spaces. We show that in the presence of Baireness fragmentability of a topological group is very close to metrizability of that group. We also show that for a compact Hausdorff space X, sigma-fragmentability of (C(X). parallel to . parallel to(infinity)) implies that the space C(p)(X: M) of all continuous functions from X into a metric space M. endowed with the topology of pointwise convergence on X, is fragmented by a metric whose topology is at least as strong as the uniform topology on C(X: M). The primary tool used is that of topological games. (C) 2011 Elsevier B.V. All rights reserved.