A question on Lie sub-group Well, definition of Lie subgroup what I know is, a Lie subgroup of a Lie group $G$ is an abstract subgroup $H$ which is an immersed submanifold via the inclusion map so that the group operations on $H$ are $C^{\infty}$.
Could any one make me understand the following with an example?
"Because a Lie subgroup is an immersed submanifold, it need not have the relative topology. In particular, the inclusion map $i:H\rightarrow G$ need not be continuous."

I can't help you understand that statement, because I don't understand it: an immersion is a differentiable map with injective derivative, and is in particular continuous. A standard example of a Lie subgroup such that the inverse of the inclusion is not continuous is the image of a line with irrational slope in the two-dimensional compact torus (quotient of the plane by $\mathbb{Z}^2$).
As Paul Garrett mentions in a comment above, this kind of example is almost certainly what Tu had in mind when he wrote the statement you quote. One reason for making the definition this way instead of with a more restrictive definition of Lie subgroup is to be flexible enough to allow Lie subgroups of a Lie group to correspond to Lie subalgebras of its Lie algebra.
