Consider the one-dimensional Lie group
plotted as a subgroup embedded (or immersed) in the two-dimensional torus group
, also an Abelian Lie group. The slope of the one-dimensional group is
, and if the slope is a rational number of the form
are relatively prime (have no common factors), then
is compact: it is a closed loop wound around the torus and joins onto itself (comes back to the group identity) when
. If, however,
is not a loop; it is noncompact and indeed isomorphic to the one-dimensional Lie group
. One can readily understand that this is true from the conditions for a closed loop: there are angles
for some integer
for another integer
, contradicting the irrationality of
. Moreover, the subgroup is dense in the torus; if it were not, then there would some open ball
(open in the group topology of
) such that
. But, since
is a group; we then see that
for every group member
. This means that, for example, the intersections of
with the loop
1. distinct (otherwise
closes on itself, gainsaying the irrationality of
); that is, there are countably infinitely many of them, and
2. spaced by at least
, which is impossible, because the loop is compact, thus has the Bolzano–Weierstrass property, and the necessary limit point of the set of intersections means that the spacings between intersections has a greatest lower bound of zero.
A theorem due to Cartan shows that any closed subgroup of a Lie group is itself a Lie group: this is the situation we have for
is rational and
is closed in the Lie group
. In this case, the topology that makes
a Lie group is the relative topology inherited by
is a topological embedding within
. However, when
is not closed in
, and the topology that must be given to it to make it a Lie group is the group topology as defined in §2.3 of , and in this case, this topology is different from the relative topology.
is no longer a topological embedding in
is irrational, but is instead an immersion in
. Some authors  use the terminology "virtual Lie subgroup" for a Lie subgroup that fails to be a topological embedding; however, it is important to realize there is nothing "virtual" about the Lie-hood of such a subgroup when the latter is given the right topology, as in §2.3 of .
This example is an excellent illustration of the Lie correspondence, as discussed in .  also proves the Lie correspondence by a similar method to that used in .
 W. Rossmann, Lie Groups: An Introduction through Linear Groups
(Oxford Graduate Texts in Mathematics
), Oxford: Oxford University Press, 2003.
 V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg, Foundations of Lie Theory and Lie Transformation Groups
, Berlin: Springer-Verlag, 1997, §2.3 "Virtual Lie Subgroups," as well as Theorem 5.4.