Groups don't normally require any structure on their members beyond what's required to make the group operator work properly. You can define a group whose values are a set of points, a set of numbers, a set of coins - very nearly anything you want.
But there are certain structured sets of values that we care about, which you can use as the objects for a group. One of those is a topological space. A topological space is just a collection of objects which have a kind of nearness/adjacency relationship between the objects in the collection. So a group on a topological space is interesting, because what it does is define symmetry on a set of values that preserves the nearness/adjacency relationships of the objects in the space.
Even more interesting, we can define a particular kind of topological space: a manifold, which is a sort of "smooth" topological space: a manifold is a topological space where the structure of the nearness/adjacency relations makes every small finite region of the space appear to be Euclidean.
So a Lie group is a group whose objects form a manifold, and whose group operations preserve the manifold structure of the nearness/adjacency relations."
http://scienceblogs.com/goodmath/2007/03/the_mapping_of_the_e8_lie_grou.php
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