I am looking for a book that have some detailed proofs of the main results around some topics listed below:
- rank (maybe with some properties of matrices with constant rank)
- linear group
- special group (or unimodular group)
- continuity and differentiation of determinants (to prove that some groups are open)
- Orthogonal group
- connected components of these matrix groups
So the approach have to be wide and include analysis (calculus), topology, algebra and some geometry (topological and differentiable manifolds) with some isomorphisms/difeomorfisms/homeomorphisms with other objects like spheres, torus or other surfaces on euclidean spaces.
Brian C. Hall's Lie Groups, Lie Algebras, and Representations devotes a significant portion of the first chapter to specific matrix Lie groups, namely the general/special linear groups, (special and complex) orthogonal groups, (special) unitary groups, Lorentz groups, symplectic groups, Heisenberg group, Euclidean and Poincare groups, etc.
A few sections later, he investigates the connected components of many of these groups. So this might address some of the points you're looking for.