(0:00) We survived Real Analysis tee shirt. (1:28) Descriptive theory day (topological definitions). (2:23) Facts about linear fractional transformations and a demonstration from the Wolfram Demonstrations project. (5:29) Topological definitions, starting with an open disk (neighborhood) centered at a point. (8:10) Interior point of a set. (9:50) Interior of a set. (10:32) Open sets (also note that the entire complex plane and the empty set are open...though I made a mistake in how I described the empty set as open). (13:03) Polygonal path (or "curve").
Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.