Consider the one-dimensional Lie group
![](HTMLImages/index.en/6.gif)
plotted as a subgroup embedded (or immersed) in the two-dimensional torus group
![](HTMLImages/index.en/7.gif)
, also an Abelian Lie group. The slope of the one-dimensional group is
![](HTMLImages/index.en/8.gif)
, and if the slope is a rational number of the form
![](HTMLImages/index.en/9.gif)
where
![](HTMLImages/index.en/10.gif)
and
![](HTMLImages/index.en/11.gif)
are relatively prime (have no common factors), then
![](HTMLImages/index.en/12.gif)
is compact: it is a closed loop wound around the torus and joins onto itself (comes back to the group identity) when
![](HTMLImages/index.en/13.gif)
. If, however,
![](HTMLImages/index.en/14.gif)
is irrational,
![](HTMLImages/index.en/15.gif)
is not a loop; it is noncompact and indeed isomorphic to the one-dimensional Lie group
![](HTMLImages/index.en/16.gif)
. One can readily understand that this is true from the conditions for a closed loop: there are angles
![](HTMLImages/index.en/17.gif)
,
![](HTMLImages/index.en/18.gif)
such that
![](HTMLImages/index.en/19.gif)
for some integer
![](HTMLImages/index.en/20.gif)
and also
![](HTMLImages/index.en/21.gif)
for another integer
![](HTMLImages/index.en/22.gif)
, whence
![](HTMLImages/index.en/23.gif)
, contradicting the irrationality of
![](HTMLImages/index.en/24.gif)
. Moreover, the subgroup is dense in the torus; if it were not, then there would some open ball
![](HTMLImages/index.en/25.gif)
(open in the group topology of
![](HTMLImages/index.en/26.gif)
) such that
![](HTMLImages/index.en/27.gif)
. But, since
![](HTMLImages/index.en/28.gif)
for any
![](HTMLImages/index.en/29.gif)
,
![](HTMLImages/index.en/30.gif)
is a group; we then see that
![](HTMLImages/index.en/31.gif)
for every group member
![](HTMLImages/index.en/32.gif)
. This means that, for example, the intersections of
![](HTMLImages/index.en/33.gif)
with the loop
![](HTMLImages/index.en/34.gif)
must be
1. distinct (otherwise
![](HTMLImages/index.en/35.gif)
closes on itself, gainsaying the irrationality of
![](HTMLImages/index.en/36.gif)
); that is, there are countably infinitely many of them, and
2. spaced by at least
![](HTMLImages/index.en/37.gif)
, 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
![](HTMLImages/index.en/38.gif)
when
![](HTMLImages/index.en/39.gif)
is rational and
![](HTMLImages/index.en/40.gif)
is closed in the Lie group
![](HTMLImages/index.en/41.gif)
. In this case, the topology that makes
![](HTMLImages/index.en/42.gif)
a Lie group is the relative topology inherited by
![](HTMLImages/index.en/43.gif)
from
![](HTMLImages/index.en/44.gif)
, and
![](HTMLImages/index.en/45.gif)
is a topological embedding within
![](HTMLImages/index.en/46.gif)
. However, when
![](HTMLImages/index.en/47.gif)
is irrational,
![](HTMLImages/index.en/48.gif)
is not closed in
![](HTMLImages/index.en/49.gif)
, 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 [1], and in this case, this topology is different from the relative topology.
![](HTMLImages/index.en/50.gif)
is no longer a topological embedding in
![](HTMLImages/index.en/51.gif)
when
![](HTMLImages/index.en/52.gif)
is irrational, but is instead an immersion in
![](HTMLImages/index.en/53.gif)
. Some authors [2] 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 [1].
This example is an excellent illustration of the Lie correspondence, as discussed in [3]. [1] also proves the Lie correspondence by a similar method to that used in [3].
[1] W. Rossmann,
Lie Groups: An Introduction through Linear Groups (
Oxford Graduate Texts in Mathematics), Oxford: Oxford University Press, 2003.
[2] 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.