Real Number Line is Non-Meager/Proof 2

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

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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Sources

• 1973: Thomas J. Jech: The Axiom of Choice ... (previous) ... (next): $1.$ Introduction: $1.4$ Problems: $5$