structural:reformatting, stylistic:style_transfer, linguistic:paraphrasing
Topological definition

A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup G of the general linear group GL (n, C) {\displaystyle \operatorname {GL} (n,\mathbb {C})} such that

(For example, a closed subgroup of GL (n, C) {\displaystyle \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:

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.