Abstract
We work in set theory without the axiom of choice: ZF . We show that the axiom BC : Compact Hausdorff spaces are Baire , is equivalent to the following axiom: Every tree has a subtree whose levels are finite , which was introduced by Blass (cf. [ 4 ]). This settles a question raised by Brunner (cf. [ 9 , p. 438]). 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 Luxemburg (cf. [ 16 ]).
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