Lecture
This lecture covers the concept of smooth functions on manifolds, starting with examples of smooth functions defined on specific manifolds. It delves into the definition of smoothness at a point on a manifold using charts, emphasizing the importance of continuity in defining smooth functions. The lecture also discusses the atlas topology associated with a set of charts, highlighting how it can lead to non-unique limits in certain cases. Furthermore, it explores the uncomfortable nature of atlas topologies and the criteria for a smooth manifold to have a Hausdorff and second-countable atlas topology. The concept of embedded submanifolds and their unique maximal atlases is also explained, along with the conditions for a function on a manifold to be smooth.