THE BAIRE CATEGORY PROPERTY AND SOME NOTIONS OF COMPACTNESS

We work in set theory without the axiom of choice: ZF. We show that th e axiom BC: Compact Hausdorff spaces are Baire, is equivalent to the f ollowing axiom: Every tree has a subtree whose levels are finite, whic h 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 C hoices is equivalent to the axiom: In a Hausdorff locally convex topol ogical vector space, convex-compact convex sets are Baire. Here convex -compact is the notion which was introduced by Luxemburg (cf. [16]).