This textbook discusses all useful topics in general topology and applications. It contains complete definitions of topics well-explained by suitable examples, explanations, and proofs throughout the book. The book studies major topics on topological spaces and continuous functions, countability, separable spaces and connectedness, compactness and the Tychonoff theorem, separation axioms, compactification, products and coproducts, embedding, metrization theorem and paracompactness, nets and filters, complete metric spaces and function spaces, uniform spaces, manifolds and their geometry, Baire spaces and dimension geometry. This book is designed as a textbook for a two-semester course in point-set topology or metric topology, as well as for a first-semester course in topology at the lower undergraduate level. The book is also accessible to junior mathematics majors who have studied multivariable calculus. The essential prerequisites for reading this book are quite minimal: not much more than a stiff course in analysis, advanced calculus, and abstract algebra.