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Valdivia compact
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Valdivia compact をテンプレートにして作成
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開始行:
*Definition [#g30845fc]
-A compact Hausdorff space X is Valdivia compact if X has a dense [[Σ-subset]].
*Property [#ae97f0db]
- 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.
++ Π_{a \in Λ} X_a is Valdivia compact.
++ 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.
++ X is Valdivia compact.
++ The space P(X) of all Radon probabilities on X, endowed with the weak* topology, is Valdivia compact.
++ The dual unit ball B_{C(X)*}, endowed with the weak* topology, is Valdivia compact.
++ The dual unit ball B_{C(X)*}, endowed with the weak* topology, has a convex symmetric dense Σ-subset.
*Reference [#g998b14c]
Kalenda, Ondřej(CZ-KARLMP-MA),''A characterization of Valdivia compact spaces'', (English summary)Collect. Math. 51 (2000), no. 1, 59--81.
終了行:
*Definition [#g30845fc]
-A compact Hausdorff space X is Valdivia compact if X has a dense [[Σ-subset]].
*Property [#ae97f0db]
- 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.
++ Π_{a \in Λ} X_a is Valdivia compact.
++ 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.
++ X is Valdivia compact.
++ The space P(X) of all Radon probabilities on X, endowed with the weak* topology, is Valdivia compact.
++ The dual unit ball B_{C(X)*}, endowed with the weak* topology, is Valdivia compact.
++ The dual unit ball B_{C(X)*}, endowed with the weak* topology, has a convex symmetric dense Σ-subset.
*Reference [#g998b14c]
Kalenda, Ondřej(CZ-KARLMP-MA),''A characterization of Valdivia compact spaces'', (English summary)Collect. Math. 51 (2000), no. 1, 59--81.
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