Local Homeomorphisms and Coverings

This lecture covers the concepts of local homeomorphisms and coverings in the context of manifolds, emphasizing the conditions under which a map between manifolds is considered a local homeomorphism or a covering. The lecture also introduces the notion of connected components and the mapping of these components under local homeomorphisms. The Riemann-Hurwitz formula is presented as a tool to analyze the behavior of non-constant holomorphic maps between compact connected Riemann surfaces. The lecture concludes with exercises on triangulations, branch points, and the construction of compact Riemann surfaces.