Editing Lie group (section)

=== Non-example ===
{{further|Linear flow on the torus}}
We now present an example of a group with an [[uncountable set|uncountable]] number of elements that is not a Lie group under a certain topology.  The group given by 
: <math>H = \left\{\left(\begin{matrix}e^{2\pi i\theta} & 0\\0 & e^{2\pi ia\theta}\end{matrix}\right) :\, \theta \in \mathbb{R}\right\} \subset \mathbb{T}^2 = \left\{\left(\begin{matrix}e^{2\pi i\theta} & 0\\0 & e^{2\pi i\phi}\end{matrix}\right) :\, \theta, \phi \in \mathbb{R}\right\},</math>
with <math>a \in \mathbb R \setminus \mathbb Q</math> a ''fixed'' [[irrational number]], is a subgroup of the [[torus]] <math>\mathbb T^2</math> that is not a Lie group when given the [[subspace topology]].<ref>{{harvnb|Rossmann|2001|loc=Chapter 2}}</ref> If we take any small [[neighborhood (mathematics)|neighborhood]] <math>U</math> of a point <math>h</math> in {{tmath|1= H }}, for example, the portion of <math>H</math> in <math>U</math> is disconnected. The group <math>H</math> winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a [[dense set|dense]] subgroup of {{tmath|1= \mathbb T^2 }}.

[[File:Irrational line on a torus.png|thumb|right|A portion of the group <math>H</math> inside {{tmath|1= \mathbb T^2 }}. Small neighborhoods of the element <math>h\in H</math> are disconnected in the subset topology on {{tmath|1= H }}]]

The group <math>H</math> can, however, be given a different topology, in which the distance between two points <math>h_1,h_2\in H</math> is defined as the length of the shortest path ''in the group'' <math>H</math> joining <math>h_1</math> to {{tmath|1= h_2 }}. In this topology, <math>H</math> is identified homeomorphically with the real line by identifying each element with the number <math>\theta</math> in the definition of {{tmath|1= H }}. With this topology, <math>H</math> is just the group of real numbers under addition and is therefore a Lie group.

The group <math>H</math> is an example of a "[[Lie group#Lie subgroup|Lie subgroup]]" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.