## 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

### Citations:

Zbl 0526.03031; Zbl 0181.40101
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