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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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