Euclidean Space without Origin is Path-Connected

Theorem

Let $n \in \Z: n \ge 2$.

Let $\R^n$ be the $n$-dimensional Euclidean space.

Let $\R^n \setminus \set {\mathbf 0}$ be $\R^n$ with the origin removed.

Then $\R^n \setminus \set {\mathbf 0}$ is path-connected.

Proof

This theorem requires a proof. In particular: Another day, this is tedious. 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