Relation To The Axiom of Choice
The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.
The restricted form of the Baire category theorem in which the complete metric space is also assumed to be separable is provable in ZF with no additional choice principles. This restricted form applies in particular to the real line, the Baire space ωω, and the Cantor space 2ω.
Read more about this topic: Baire Category Theorem
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