Editing Lie group (section)

=== Topological definition ===
A Lie group can be defined as a ([[Hausdorff space|Hausdorff]]) [[topological group]] that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds.<ref>{{harvnb|Kobayashi|Oshima|2005|loc=Definition 5.3}}</ref> First, we define an '''immersely linear Lie group''' to be a subgroup ''G'' of the general linear group <math>\operatorname{GL}(n, \mathbb{C})</math> such that
# for some neighborhood ''V'' of the identity element ''e'' in ''G'', the topology on ''V'' is the [[subspace topology]] of <math>\operatorname{GL}(n, \mathbb{C})</math> and ''V'' is closed in {{tmath|1= \operatorname{GL}(n, \mathbb{C}) }}.
# ''G'' has at most [[countable set|countably many]] connected components.
(For example, a closed subgroup of {{tmath|1= \operatorname{GL}(n, \mathbb{C}) }}; that is, a matrix Lie group satisfies the above conditions.)

Then a ''Lie group'' is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:
# Given a Lie group ''G'' in the usual manifold sense, the [[Lie group–Lie algebra correspondence]] (or a version of [[Lie's third theorem]]) constructs an immersed Lie subgroup <math>G' \subset \operatorname{GL}(n, \mathbb{C})</math> such that <math>G, G'</math> share the same Lie algebra; thus, they are locally isomorphic. Hence, <math>G</math> satisfies the above topological definition.
# Conversely, let <math>G</math> be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group <math>G'</math> that is locally isomorphic to {{tmath|1= G }}. Then, by a version of the [[closed subgroup theorem]], <math>G'</math> is a [[real-analytic manifold]] and then, through the local isomorphism, ''G'' acquires a structure of a manifold near the identity element. One then shows that the group law on ''G'' can be given by formal [[power series]];{{efn|This is the statement that a Lie group is a [[formal Lie group]]. For the latter concept, see Bruhat.<ref>{{Cite web |first=F. |last=Bruhat |url=http://www.math.tifr.res.in/~publ/ln/tifr14.pdf |title=Lectures on Lie Groups and Representations of Locally Compact Groups |date=1958 |publisher=Tata Institute of Fundamental Research, Bombay}}</ref>}} so the group operations are real-analytic and <math>G</math> itself is a real-analytic manifold.

The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, ''the topology of a Lie group'' together with the group law determines the geometry of the group.