Real Number Line is Non-Meager/Proof 1

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Then $\struct {\R, \tau_d}$ is non-meager.

Proof

We have that the Real Number Line is Complete Metric Space.

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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology: $1$