Set Theory & Analysis

On Asplund spaces C_k(X)

 We continue a study  about  two classes of Delta-spaces and Delta_1-spaces  which  provide  extensions of the family of Delta-sets and lambda-sets, respectively,  beyond the separable metrizable spaces. A question  about a characterisation of a Tychonoff space X as being a Delta_1-space  by a  suitable analytic property of  C_k(X) or its strong dual is studied. We show   that if X is omega-bounded (or Cech-complete pseudocompact), then X is scattered iff  X is a Delta_1-space. Consequently,  a compact space X is a Delta_1-space iff X is scattered. On the other hand, we show that if X is omega-bounded (or Cech-complete pseudocompact), then X is scattered iff  C_k(X) is Asplund, i.e. every separable Banach subspace of C_k(X) has separable dual. This   extends the Amir-Lindestrauss theorem.