$X$ is contractible if the identity map $i_X : X \to X$ is homotopic to a constant map. Prove $\mathbb{R} ^n$ is contractible for any $n \geq 1.$ Is any contractible space is path connected?
I am not sure how to show $\mathbb{R} ^n$ is contractible for any $n \geq 1,$ but I will attempt to show any contractible space is path connected. Suppose $X$ is contractible. Then there exists a homotopy $H$ such that $H(x,0) = x$ and $H(x,1) = c$ for all $x \in X$ and for some fixed $c \in X$. Then every point of $X$ is connected to $c$ by a path, and hence any two points $x_1$ and $x_2$ of $X$ can be joined by a path through $c$, completing the proof.
Best Answer
For $\mathbb{R}^n$ use $H(x,t)=(1-t)x$ for $(x,t)\in\mathbb{R}^n\times[0,1]$
You have that $H(x,0)=x$ and $H(x,1)=0$.