Baire Category Theorem/Corollary

Corollary to Baire Category Theorem

The real number line $\R$ with the usual (Euclidean) metric is non-meager.

Proof 1

From Real Number Line is Complete Metric Space, $\R$ is a complete metric space.

From the Baire Category Theorem, a complete metric space is a Baire space.

The result follows from Baire Space is Non-Meager.

$\blacksquare$

Axiom of Dependent Choice

This proof depends on the Axiom of Dependent Choice, by way of Baire Category Theorem.

Although not as strong as the Axiom of Choice, the Axiom of Dependent Choice is similarly independent of the Zermelo-Fraenkel axioms.

The consensus in conventional mathematics is that it is true and that it should be accepted.

Proof 2

This proof does not use the Axiom of Dependent Choice, as it uses intrinsic properties of the real numbers that do not necessarily hold for the general complete metric space.

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Source of Name

This entry was named for René-Louis Baire.