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

where

and

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 irrational,

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

,

such that

for some integer

and also

for another integer

, whence

, 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

for any

,

is a group; we then see that

for every group member

. This means that, for example, the intersections of

with the loop

must be

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

when

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

from

, and

is a topological embedding within

. However, when

is irrational,

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 [1], and in this case, this topology is different from the relative topology.

is no longer a topological embedding in

when

is irrational, but is instead an immersion in

. 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.