Continuous Functions on Compact Totally Ordered Spaces

The Banach space C(K) of continuous functions on K with its pointwise topology is shown to be σ-fragmented by pointwise closed sets, when K is a compact totally ordered set, when K is a dendron, and when K is a compact Hausdorff space with a closed crossfree subbase for its topology. In some special instances, C(K) will have an equivalent locally uniformly convex norm.