Locally compact groups, residual Lie groups, and varieties generated by Lie groups

The concept of approximating in various ways locally compact groups by Lie groups is surveyed with emphasis on pro-Lie groups and locally compact residual Lie groups. All members of the variety of Hausdorff groups generated by the class of all finite dimensional real Lie groups are residual Lie groups. Conversely, we show that every locally compact member of this variety is a pro-Lie group. For every locally compact residual Lie group we construct several better behaved residual Lie groups into which it is equidimensionally immersed. We use such a construction to prove that for a locally compact residual Lie group G the component factor group GG0 is residually discrete.