What topics will be covered in the course?

Review of topology, including open and closed sets, continuous functions between topological spaces, bases of a topological space and axioms of countability, separation axioms, connectivity and compactness, quotient spaces, locally compact spaces, fundamental groups, and covering spaces.
Review of important theorems in vector calculus and differential equations, including the inverse function theorem, implicit function theorem, change of variables (integration), and existence and uniqueness theorem.
Formalization of the notion of a space that locally looks like Rn but globally can have a very different geometry intrinsically.
Construction of certain constructions in these spaces that will allow us to do calculus on them (such as differentiation and integration).
Applications to physics as necessary.