----- > [!proposition] Proposition. ([[past one dimension, punctured euclidean space is path-connected]]) > $\mathbb{R}^{k} - \{ \b 0 \}$ is [[path-connected]] provided that $k \geq 2$. > [!proposition] Corollary. > $\mathbb{R}^{k}$ is not [[homeomorphism|homeomorphic]] to $\mathbb{R}$ for $k \geq 2$. > \ > Indeed, the image of the ([[path-connected]], [[path-connected versus connected|hence connected]] subset) $\mathbb{R}^{k}-\{ \b 0 \}$ under any prospective [[homeomorphism]] $f:\mathbb{R}^{k} \to \mathbb{R}$ would need to be a [[connected]] subset of $\mathbb{R}$, but, using the fact that $f$ is a [[bijection]], $f(\mathbb{R}^{k} \cut \{ \b 0 \} )=f(\mathbb{R}^{k}) \cut \{ f(0) \}=\mathbb{R} - f(\b 0),$ which is not a [[connected]] subset of $\mathbb{R}$. ^eeae4d > [!proof]- Proof. ([[past one dimension, punctured euclidean space is path-connected]]) > Fix $\b p,\b q \in \mathbb{R}^{k}-\{ \b 0 \}$. > Define $f:[0,1] \to \mathbb{R}^{k}$ to be the [[line]] connecting $\b p$ and $\b q$, $f(t)=t \b p + (1-t) \b q$. If $f$ is never $\b 0$ then we have successfully witnessed a [[parameterized curve]] from $\b p$ to $\b q$. If $f$ is zero at some point, we just take a point $\b v$ not one; the line from $\b p$ to $\b q$ and define $\begin{align} f_{1}: & [0, 1] \to \mathbb{R}^{k} , \ \ f_{1}(t):= t\b v + (1-t) \b p \\ f_{2} : & [1, 2] \to \mathbb{R}^{k}, \ \ f_{2}(t) := (t-1) \b q + (2-t) \b v; \end{align}$ note that since $f_{1}(1) = \b v = f_{2}(1)$ [[the pasting lemma]] guarantees that the 'broken line' function $f_{3}:[0,2] \to \mathbb{R}^{k}$ given by $f_{3}(t) = \begin{cases} f_{1}(t) & t \in [0,1] \\ f_{2}(t) & t \in [1,2] \end{cases}$ is [[continuous]]. Since $[0,2]$ is [[homeomorphism]] to $[0,1]$, we hence know that there exists a path with the same trace as $f_{3}$ from $\b p$ to $\b q$, and this path never touches $\b 0$ by construction. ^5740e2 ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```