Real Number Line is Non-Meager

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.

$\blacksquare$

Sources

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