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Let $G$ be a locally compact group and let $(x_{\alpha})$ be a convergent net, say to $x$, in $G$. Is it possible to construct a compact subset $K$ of $G$ which contains each $x_{\alpha}$ and $x$?

roo
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  • Maybe I don't understand; why not take $K = G$? Or the closure of $(x_\alpha)$. Any closed subset of a compact space is again compact; did you want $K$ to be a strict subgroup of $G$? – felipeh May 29 '14 at 21:19
  • $G$ need not be compact. But going back to the closure of $(x_{\alpha})$. Need this be compact? – roo May 29 '14 at 21:29
  • Oh sorry I misread. – felipeh May 29 '14 at 22:11

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Not necessarily. Consider $G=\Bbb R,$ $x=0,$ and the net $(0,\infty)\to\Bbb R$--with $(0,\infty)$ directed in increasing order--given by $x_\alpha=\frac1\alpha.$ Any subset of $\Bbb R$ containing every $x_\alpha$ must fail to be bounded, so fail to be compact.

Cameron Buie
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  • The context of my problem was I needed to be working in a compact set to apply a result. I think the solution (to my unposted problem) is to take a relatively compact neighborhood of $x$, and work with the subnet of terms that lie within it. – roo May 29 '14 at 21:46