Unit Interval is Path-Connected in Real Numbers

Theorem

Let $\R$ be the real number line with the usual (Euclidean} metric.

The closed unit interval $\mathbf I = \closedint 0 1$ is a path-connected metric subspace of $\R$.

Proof

Follows directly from Subset of Real Numbers is Path-Connected iff Interval.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Path-Connectedness