# The punctured euclidean space is simply connected (dim>2)

• Jan 12th 2011, 04:46 AM
CSM
The punctured euclidean space is simply connected (dim>2)
I'm looking for a proof of the statement that $\displaystyle \mathbb{R}^n\backslash \{0\}$ is simply connected.

I've already got a proof, but I'm trying to find another one (one that uses the fact that a product of simply connected space is again simply connected).
Proof: Any loop $\displaystyle \gamma$ in $\displaystyle \mathbb{R}^n\backslash \{0\}$
can be homotopied to a path on the unit sphere $\displaystyle S^{n-1}$ by $\displaystyle \gamma_t(s)=\gamma(s)/||\gamma(s)||^t$. Paths on the sphere $\displaystyle S^{n-1}$ can be homotopied to a point for $\displaystyle n>2$
• Jan 13th 2011, 11:00 PM
Drexel28
Quote:

Originally Posted by CSM
I'm looking for a proof of the statement that $\displaystyle \mathbb{R}^n\backslash \{0\}$ is simply connected.

I've already got a proof, but I'm trying to find another one (one that uses the fact that a product of simply connected space is again simply connected).
Proof: Any loop $\displaystyle \gamma$ in $\displaystyle \mathbb{R}^n\backslash \{0\}$
can be homotopied to a path on the unit sphere $\displaystyle S^{n-1}$ by $\displaystyle \gamma_t(s)=\gamma(s)/||\gamma(s)||^t$. Paths on the sphere $\displaystyle S^{n-1}$ can be homotopied to a point for $\displaystyle n>2$

What other proof could you hope for besides using the fundamental group?