Euclidean Space without Origin is Path-Connected
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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
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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