Sorry but this question may be more appropriate for stackexchange for the history of science and mathematics. I was wanting to learn a little theory on Lie Groups and having a background in finite group theory I thought I would start with smooth manifolds of which I know nothing about.
Wikipedia says a smooth manifold is an area of topology but I looked in a Schaum's outline of General Topology at the local library and there is no mention of a manifold. It reminded me of Rudin's Real Analysis but I did not see any Dedekind cuts to build the real numbers.
What branch do smooth manifolds belong to in the context of Lie Groups? I am wondering how to approach this being I am not familiar with the terrain .
Differential Geometry is probably the field you're looking for. Check out Lee's "Introduction to Smooth Manifolds". He lists the prerequisites as "general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis." He also states that "It is a natural sequel to my earlier book on topological manifolds." Here, he is referring to his book "Introduction to Topological Manifolds". This earlier book is noted as being accessible to any student who has successfully completed a good undergraduate mathematics curriculum.