List of topics

Basic frameworks: metric spaces and topological spaces; what do these structures represent?

Basis for topology: what are examples of bases? When do they become convenient?

Product topology: what can be packaged in this setting?

Continuous maps: how are they defined in the framework of topological spaces? Do bases simply this?

Topology on a space of continuous maps: what are sensible topologies (with some extra assumptions on domain / target)?

Quotient topology: how can we use this idea?

Connectedness, path-connectedness

Compactness: what are examples and non-examples?

Homotopy, fundamental group: what do they represent? How can we compute the fundamental group of a space? What can be said from the nontriviality of the fundamental group of the circle?