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The Baire category property and some notions of compactness. (English) Zbl 0922.03070

We work in set theory without the axiom of choice: \({\mathbf {ZF}}\). We show that the axiom \({\mathbf {BC}}\): “Compact Hausdorff spaces are Baire” is equivalent to the following axiom: “Every tree has a subtree whose levels are finite”. This settles a question raised by N. Brunner [Z. Math. Logik Grundlagen Math. 29, 435-443 (1983; Zbl 0526.03031)].
We also show that the axiom of dependent choices is equivalent to the axiom: “In a Hausdorff locally convex topological vector space, convex-compact convex sets are Baire”. Here convex-compact is the notion which was introduced by W. A. J. Luxemburg [Appl. Model Theory Algebra, Anal., Probab. 1967, 123-137 (1969; Zbl 0181.40101)].
We also prove the following result in \({\mathbf {ZF}}\): For every set \(I\), the closed unit ball in \(l^1 (I)\) is effectively convex-compact in the *-weak topology \(\sigma (l^1 (I), l^0 (I))\).

MSC:

03E25 Axiom of choice and related propositions
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
54D30 Compactness
54E52 Baire category, Baire spaces
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