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Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit if the groups are totally disconnected, but this hypothesis is too constraining.)

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I believe for projective limits it is always compact: its is a closed subgroup of a product of compact spaces. –  Benjamin Steinberg Nov 30 '12 at 18:59
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Inductive limits of compact things will almost never be compact. –  Benjamin Steinberg Nov 30 '12 at 19:04
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Of course, one should specify the category in which one is taking (co)limits; I guess you mean in the category of topological groups? –  Todd Trimble Nov 30 '12 at 19:22
    
Right. If I'm not mistaken, the category of compact Hausdorff topological groups has all limits and colimits, but for most of the colimits it is necessary to take the Bohr compactification of the colimit in topological groups. –  Qiaochu Yuan Nov 30 '12 at 19:39
    
Thank you all so far for your very quick reaction. Benjamin Steinberg, yes, with Tychonov's theorem; this is what I was suspecting too, I was just needing some confirmation. Could you please comment more on the inductive limit, since this is what interests me more? Qiaochu Yuan, could you give any bibliographical hint? I have no experience on the subject. Todd Trimble: yes. –  Alex M. Nov 30 '12 at 20:09
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