Smooth sets and functions: Smooth functions, topology, and manifolds

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.