In the second part of the course, we first covered basic notions of topology: open / closed / compact subset, in the framework of metric space (Rudin Ch2). Then, we studied the notion of continuous maps between metric spaces(Rudin Ch4). We give three characterizations of continuous maps, using $\epsilon-\delta$ language, using convergent sequences, and the most general notion: “preimage of open set is open”. Continuous maps sends compact set to compact set, and sends connected set to connected set. We also discussed the notion of uniform continuity (property of a single function), and the notion of uniform convergence (property for a sequence of functions), not to be confused.

The textbook to read is Rudin Ch 2, Ch 4, and Ch 7's first three sections. Ross's section 13-15, 17-22, 24, 25. We have mainly used Rudin's exercise for the homework, now for review purpose, you can take a look at Ross's exercises.