Introduction to Set Theory and Topology

Book excerpt: Introduction to Set Theory and Topology describes the fundamental concepts of set theory and topology as well as its applicability to analysis, geometry, and other branches of mathematics, including algebra and probability theory. Concepts such as inverse limit, lattice, ideal, filter, commutative diagram, quotient-spaces, completely regular spaces, quasicomponents, and cartesian products of topological spaces are considered. This volume consists of 21 chapters organized into two sections and begins with an introduction to set theory, with emphasis on the propositional calculus and its application to propositions each having one of two logical values, 0 and 1. Operations on sets which are analogous to arithmetic operations are also discussed. The chapters that follow focus on the mapping concept, the power of a set, operations on cardinal numbers, order relations, and well ordering. The section on topology explores metric and topological spaces, continuous mappings, cartesian products, and other spaces such as spaces with a countable base, complete spaces, compact spaces, and connected spaces. The concept of dimension, simplexes and their properties, and cuttings of the plane are also analyzed. This book is intended for students and teachers of mathematics.