Valdivia compact

Definition

• A compact Hausdorff space X is Valdivia compact if X has a dense Σ-subset.

Property

• Let f:X→Y be a continuous open surjection between compact Hausdorff spaces. Suppose, moreover, that Y has a dense set of G_δ points. If X is super-Valdivia , then Y is Corson.

• Let X be a Valdivia compact space with a dense set of G_δ points?. Then every continusous open image of X is Valdivia.
• Let X and Y be nonempty compact Hausdorff spaces such that X has a dense set of G_δ points and X×Y is Valdivia compact, then so are X and Y.
• Let X_a, a \in Λ be an arbitrary family of nonempty compact Hausdorff spaces such that each X_a has a dense subset of G_δ points. Then the follwing two conditions are equivalent.
  1. Π_{a \in Λ} X_a is Valdivia compact.
  2. X_a is Valdivia compact for every a \in Λ.
• Let X be a compact space with a dense set of G_δpoints. Then the following assertions are equivalent.
  1. X is Valdivia compact.
  2. The space P(X) of all Radon probabilities on X, endowed with the weak* topology, is Valdivia compact.
  3. The dual unit ball B_{C(X)*}, endowed with the weak* topology, is Valdivia compact.
  4. The dual unit ball B_{C(X)*}, endowed with the weak* topology, has a convex symmetric dense Σ-subset.

Reference

Kalenda, Ondřej(CZ-KARLMP-MA),A characterization of Valdivia compact spaces, (English summary)Collect. Math. 51 (2000), no. 1, 59--81.