Subset of Real Numbers is Path-Connected iff Interval

Theorem

Let $\R$ be the real number line considered as an Euclidean space.

Let $S \subseteq \R$ be a subset of $\R$.


Then $S$ is a path-connected metric subspace of $\R$ if and only if $S$ is a real interval.


Proof

Necessary Condition

Let $S$ be a path-connected metric subspace of $\R$.

From Path-Connected Space is Connected, it follows that $S$ is connected.

From Subset of Real Numbers is Interval iff Connected, it follows that $S$ is a real interval.

$\Box$


Sufficient Condition

Let $S$ be a real interval.


This theorem requires a proof.
In particular: We have Connected Open Subset of Euclidean Space is Path-Connected but need to show that this still holds when $S$ is closed.
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Sources

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