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Valdivia compact

Last-modified: 2010-12-18 (土) 21:18:42

Definition Edit

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

Property Edit

  • 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 Edit

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