TITLE: Non-Lie Subgroups
QUESTION [12 upvotes]: A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof).
This restriction, that it be a Lie subgroup, allows for a lot of poorly-behaved subgroups, for example a line with irrational slope on a torus.  This subgroup comes from a perfectly fine immersion of the Lie group $\mathbb{R}$, but it's not closed in the induced topology of the torus.  
As an example of something that's not a Lie subgroup, let $G= \mathbb{R}$, consider an uncountable set of $\mathbb{Q}$-independent points, none of which are rational, and consider the subgroup they generate.  If this were a Lie subgroup it would be the image of an uncountable discrete space (there can't be anything $1$-dimensional, since we left out the rationals), which wouldn't be second countable, hence not a manifold and not a Lie group.  
This seems like a pretty contrived example, and I suspect there is more content to 'being a Lie subgroup' than having countably many components.  However, I can't seem to pin down something that would illustrate this.  Can anyone give me an example of a connected subgroup of a Lie group that is not a Lie subgroup?

REPLY [4 votes]: Apparently, there is no consensus in the literature on how to define a Lie subgroup of a Lie group. One should be careful not to mix different definitions. The result of Yamabe (1950) states that every arcwise connected subgroup of a real Lie group is Lie subgroup (in the sense of Yamabe). Bourbaki uses the notion of an integral subgroup (Lie Groups and Lie Algebras, Chapter III, §8, Exercise 4). Onishchik/Vinberg use the notion of a virtual Lie subgroup (Foundations of Lie theory in Lie groups and Lie algebras I, p. 39, Theorem 2.4). Note that Onishchik/Vinberg also use the (stronger) notion of a Lie subgroup and present a counterexample not fulfilling their definition on page 14. For more on the technical details one can refer to Bourbaki (Chapter III, §6 and §8) or Onishchik/Vinberg.