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
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.