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The statement is:
The proof of the Baire category theorem uses the axiom of choice; in fact, the Baire category theorem is logically equivalent to a weaker version of the axiom of choice called the axiom of dependent choice.
The Baire category theorem is used in the proof of the open mapping theorem and the uniform boundedness principle. It also gives a quick proof that the reals are uncountable (since the reals are a complete metric space, and hence cannot be the countable union of points).
R. Baire (1899), Sur les fonctions de variables réelles, Annali di Mat. Ser. 3 3, 1--123.
Topology General topologyIn mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. It grew out of a number of areas, such as the detailed study of sets of points (as Functional analysis Theorems