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Thread: left-invariant complete metric on S_\infty

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    left-invariant complete metric on S_\infty

    Let $\displaystyle S_{\infty}$ be the group of all permutations of $\displaystyle \mathbb{N}$, viewed as a subspace of Baire space $\displaystyle \mathcal{N}$.
    How do I prove that there is no left-invariant complete metric on $\displaystyle S_{\infty}$?

    A metric is left-invariant if $\displaystyle d(xy, xz)=d(y, z)$, for all x, y, z.

    Thank you.
    Last edited by harriette; Jan 16th 2010 at 12:37 PM.
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