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Valdivia compact

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

Definition

• A compact Hausdorff space X is Valdivia compact if X has a dense ¦²-subset.

Property

• Let f:X¢ªY be a continuous open surjection between compact Hausdorff spaces. Suppose, moreover, that Y has a dense set of G_¦Ä points. If X is super-Valdivia , then Y is Corson.
• Let X be a Valdivia compact space with a dense set of G_¦Ä 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_¦Ä points and X¡ßY is Valdivia compact, then so are X and Y.
• Let X_a, a \in ¦« be an arbitrary family of nonempty compact Hausdorff spaces such that each X_a has a dense subset of G_¦Ä points. Then the follwing two conditions are equivalent.
1. ¦°_{a \in ¦«} X_a is Valdivia compact.
2. X_a is Valdivia compact for every a \in ¦«.
• Let X be a compact space with a dense set of G_¦Äpoints. 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 ¦²-subset.

Reference

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