Complex Analysis, Video #26 (Complex Arithmetic, Part 26.
Main Topics: definition of an open disk ("circular neighborhood"), interior point, and open set. Example: the open right half plane is an open set. Visualize why it is with Mathematica.
Review the definition of an open disk of radius "rho" centered at z0 and the set-builder notation used to define it. It's also called a circular neighborhood (I spelled it wrong in the video) of radius "rho" centered at z0. Define "interior point" of a set S. Discuss the visual interpretation of an interior point. Define a set S to be open if every point of S is an interior point of S. Use RegionPlot to illustrate for the (open) "right half plane". Show how to find a radius that proves that 2+i, 1+i, 0.5+i, and 0.1+i are interior point of the right half plane. Use Manipulate and Locator on Mma to make an animation that shows how the radius of the disk needs to decrease as we approach the boundary (in order to prove points are interior points). If we included the imaginary axis to create the closed right half plane, it would no longer be an open set because the points on that boundary would not be interior points of the set.