Editing Baire category theorem (section)

==Relation to the axiom of choice==
The proof of '''BCT1''' for arbitrary complete metric spaces requires some form of the [[axiom of choice]]; and in fact BCT1 is equivalent over [[Zermelo–Fraenkel set theory|ZF]] to the [[axiom of dependent choice]], a weak form of the axiom of choice.<ref>{{cite journal|last=Blair|first=Charles E.|title=The Baire category theorem implies the principle of dependent choices|journal=Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys.|year=1977|volume=25|issue=10|pages=933–934}}</ref>

A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be [[Separable space|separable]], is provable in ZF with no additional choice principles.{{sfn|Levy|2002|p=212}}
This restricted form applies in particular to the [[real line]], the [[Baire space (set theory)|Baire space]] <math>\omega^{\omega},</math> the [[Cantor space]] <math>2^{\omega},</math> and a separable [[Hilbert space]] such as the [[Lp space|<math>L^p</math>-space]] <math>L^2(\R^n)</math>.