Lie group G is clearly $C^{\infty}$-manifold and I know the conditions that a set is allowed to be (topological) $C^{\infty}$- manifold : Hausdorff space, locally Euclidean, and second countable basis.(Of course, differential structure should be given...) Thus, I have thought that (every) Lie group G has second countable basis... However, I feel confused when reading the following exercise. (Warner's textbook)
Prove that connected Lie group is automatically second countable. That is, the assumption of second countability in the definition of connected Lie group is redundant. .....(♣)
Even though the Lie group is connected or not, as I mentioned, Lie group has second countable basis because of the definition of (topological) manifold. If (♣) is true, disconnected Lie group would not be second countable?(Apparently, this statement is strange...) what's wrong in my thought?