I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to this question. Has any proof of this assertion been published?
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asked Feb 15, 2017 at 16:59
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2$\begingroup$ Try the following from the sister site: math.stackexchange.com/questions/1368775/… $\endgroup$– Ben McKayCommented Feb 15, 2017 at 17:06
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$\begingroup$ A connected locally compact group is $\sigma$-compact (it coincides with the group hull of any compact neighborhood if 1), so Lindelof, and being metrizable is second-countable. $\endgroup$– Taras BanakhCommented Mar 9, 2017 at 18:06
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1 Answer
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To summarize the easy argument given in MathSE by Cronus: if $V$ is a neighborhood of the unit homeomorphic to an open ball, $(B_n)$ a countable basis of $V$ and $D$ a dense countable subset of $V$, then the $(g_1\dots g_kB_n)$ when $k,n$ range over integers and $g_i$ over $D$, form a countable basis for $G$.
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$\begingroup$ This works if the Lie group is separable. Some people put separable into the definition of a manifold, but you don't have to. So the question is whether being a Lie group implies that the underlying manifold is not too big. In particular, is there a Lie group structure on the long line? $\endgroup$ Commented Feb 16, 2017 at 12:05
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$\begingroup$ @Jan-ChristophSchlage-Puchta. No, this always works and separable is a consequence of this sketched proof. Actually it's even simpler: $G$ is the union of all $g_1\dots g_kV$ when $k$ ranges over integers and $g_i$ over $D$; this is a countable union of separable subsets.. $\endgroup$– YCorCommented Feb 16, 2017 at 12:58