Mathematical Vignettes III Non–Euclidean Geometry, Topology and Complex Analysis

This book offers in-depth summaries of three topics, i.e., non-Euclidean geometry, topology (including surfaces and metric spaces) and complex analysis. My aim is to introduce readers to each subject, providing ample references for further exploration.
The section on non-Euclidean geometry provides a brief overview of Euclid's parallel postulate and how that led to the eventual development of non-Euclidean geometry. We then go on to cover hyperbolic and elliptic geometries. The prerequisite for this section is a good understanding of Euclidean geometry.
The topology section begins with an examination of surfaces, followed by a discussion on metric spaces, which smoothly transitions into topology as a generalization of these concepts. Topology, a more generalized and abstract form of geometry, is discussed at a level appropriate for upper-level college mathematics students.
The last section covers complex analysis. We start with complex numbers and then continue with complex functions. From there, we discuss continuity, differentiation and integration of complex functions. The concepts of complex series, singularities and residues are also presented. The section is written with the assumption that the reader is familiar with calculus.