# 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."

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That statement is not correct. What they probably mean is that the map $H\to G$ may not be a homeomorphism onto its image. – Eric O. Korman Aug 16 '12 at 0:15
@Eric its a copied statement from the book " Introduction to manifolds" by Loring W Tu, page-152. – La Belle Noiseuse Aug 16 '12 at 0:17
Steve's example illustrates the point of that remark, whatever words are used. And, sometimes, textbooks are "wrong" in some logical sense that is not terribly relevant. Again, the essence of the potential problem already occurs in simple cases, as in the irrational winding on torus in Steve's example. – paul garrett Aug 16 '12 at 0:35

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 $\mathbb{Z}^2$).