Fragmentability of groups and metric-valued function spaces

Abstract

Let ( X , τ ) be a topological space and let ρ be a metric defined on X. We shall say that ( X , τ ) is fragmented by ρ if whenever ε > 0 and A is a nonempty subset of X there is a τ-open set U such that U ∩ A ≠ ∅ and ρ − diam ( U ∩ A ) < ε . In this paper we consider the notion of fragmentability, and its generalisation σ-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, σ-fragmentability of ( C ( X ) , ‖ ⋅ ‖ ∞ ) 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.