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Math Help - The punctured euclidean space is simply connected (dim>2)

  1. #1
    CSM
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    The punctured euclidean space is simply connected (dim>2)

    I'm looking for a proof of the statement that \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 \gamma in \mathbb{R}^n\backslash \{0\}
    can be homotopied to a path on the unit sphere S^{n-1} by \gamma_t(s)=\gamma(s)/||\gamma(s)||^t. Paths on the sphere S^{n-1} can be homotopied to a point for n>2
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by CSM View Post
    I'm looking for a proof of the statement that \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 \gamma in \mathbb{R}^n\backslash \{0\}
    can be homotopied to a path on the unit sphere S^{n-1} by \gamma_t(s)=\gamma(s)/||\gamma(s)||^t. Paths on the sphere S^{n-1} can be homotopied to a point for n>2
    What other proof could you hope for besides using the fundamental group?
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