# Introduction to Set Theory and Topology

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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.

## Table of Contents

Foreword to the First English Edition

Foreword to the Second English Edition

Part I Set Theory

Introduction to Part I

I. Propositional Calculus

§ 1. The Disjunction and Conjunction of Propositions

§ 2. Negation

§ 3. Implication

Exercises

II. Algebra of Sets. Finite Operations

§ 1. Operations on Sets

§ 2. Inter-Relationship with the Propositional Calculus

§ 3. Inclusion

§ 4. Space. Complement of a Set

§ 5. The axiomatics of the Algebra of Sets

§ 6. Boolean Algebra. Lattices

§ 7. Ideals and Filters

Exercises

III. Propositional Functions. Cartesian Products

§ 1. The Operation {x: φ(x)}

§ 2. Quantifiers

§ 3. Ordered Pairs

§ 4. Cartesian Product

§ 5. Propositional Functions of Two Variables. Relations

§ 6. Cartesian Products of n Sets. Propositional Functions of n Variables

§ 7. On the Axiomatics of Set Theory

Exercises

IV. The Mapping Concept. Infinite Operations. Families of Sets

§ 1. The Mapping Concept

§ 2. Set-Valued Mappings

§ 3. The Mapping Fx = {y: φ(x,y)}

§ 4. Images and Inverse Images Determined by a Mapping

§ 5. The Operations ∪R and ∩R. Covers

§ 6. Additive and Multiplicative Families of Sets

§ 7. Borel Families of Sets

§ 8. Generalized Cartesian Products

Exercises

V. The Concept of the Power of a Set. Countable Sets

§ 1. One-to-one Mappings

§ 2. Power of a Set

§ 3. Countable Sets

Exercises

VI. Operations on Cardinal Numbers. The Numbers a and c

§ 1. Addition and Multiplication

§ 2. Exponentiation

§ 3. Inequalities for Cardinal Numbers

§ 4. Properties of the Number c

Exercises

VII. Order Relations

§ 1. Definitions

§ 2. Similarity. Order Types

§ 3. Dense Ordering

§ 4. Continuous Ordering

§ 5. Inverse Systems, Inverse Limits

Exercises

VIII. Well Ordering

§ 1. Well Ordering

§ 2. Theorem on Transfinite Induction

§ 3. Theorems on the Comparison of Ordinal Numbers

§ 4. Sets of Ordinal Numbers

§ 5. The Number Ω

§ 6. The Arithmetic of Ordinal Numbers

§ 7. The Well-Ordering Theorem

§ 8. Definitions by Transfinite Induction

Exercises

Part II Topology

Introduction to Part II

IX. Metric Spaces. Euclidean Spaces

§ 1. Metric Spaces

§ 2. Diameter of a Set. Bounded Spaces. Bounded Mappings

§ 3. The Hubert Cube

§ 4. Convergence of a Sequence of Points

§ 5. Properties of the Limit

§ 6. Limit in the Cartesian Product

§ 7. Uniform Convergence

Exercises

X. Topological Spaces

§ 1. Definition. Closure Axioms

§ 2. Relations to Metric Spaces

§ 3. Further Algebraic Properties of the Closure Operation

§ 4. Closed Sets. Open Sets

§ 5. Operations on Closed Sets and Open Sets

§ 6. Interior Points. Neighborhoods

§ 7. The Concept of open Set as the Primitive Term of the Notion of Topological Space

§ 8. Base and Subbase

§ 9. Relativization. Subspaces

§ 10. Comparison of Topologies

§ 11. Cover of a Space

Exercises

XI. Basic Topological Concepts

§ 1. Borel Sets

§ 2. Dense Sets and boundary Sets

§ 3. T1-spaces, T2-spaces

§ 4. Regular Spaces, Normal Spaces

§ 5. Accumulation Points. Isolated Points

§ 6. The Derived Set

§ 7. Sets Dense in Themselves

Exercises

XII. Continuous Mappings

§ 1. Continuity

§ 2. Homeomorphisms

§ 3. Case of Metric Spaces

§ 4. Distance of a Point from a Set. Normality of Metric Spaces

§ 5. Extension of Continuous Functions. Tietze Theorem

§ 6. Completely Regular Spaces

Exercises

XIII. Cartesian Products

§ 1. Cartesian Product X x Y of Topological Spaces

§ 2. Projections and Continuous Mappings

§ 3. Invariants of Cartesian Multiplication

§ 4. Diagonal

§ 5. Generalized Cartesian Products

§ 6. XT Considered as a Topological Space. The Cube JT

§ 7. Cartesian Products of Metric Spaces

Exercises

XIV. Spaces with a Countable Base

§ 1. General Properties

§ 2. Separable Spaces

§ 3. Theorems on Cardinality in Spaces with Countable Bases

§ 4. Imbedding in the Hubert Cube

§ 5. Condensation Points. The Cantor-Bendixson Theorem

Exercises

XV. Complete Spaces

§ 1. Complete Spaces

§ 2. Cantor Theorem

§ 3. Baire Theorem

§ 4. Extension of a Metric Space to a Complete Space

Exercises

XVI. Compact Spaces

§ 1. Definition

§ 2. Fundamental Properties of Compact Spaces

§ 3. Cartesian Products

§ 4. Compactification of Completely Regular Spaces

§ 5. Compact Metric Spaces

§ 6. The Topology of Uniform Convergence of YX

§ 7. The Compact-open Topology of YX

§ 8. The Cantor Discontinuum

§ 9. Continuous Mappings of the Cantor Discontinuum

Exercises

XVII. Connected Spaces

§ 1. Definition. Separated Sets

§ 2. Properties of Connected Spaces

§ 3. Components

§ 4. Cartesian Products of Connected Spaces

§ 5. Continua

§ 6. Properties of Continua

Exercises

XVIII. Locally Connected Spaces

§ 1. Definitions and Examples

§ 2. Properties of Locally Connected Spaces

§ 3. Locally Connected Continua

§ 4. Arcs. Arcwise Connectedness

§ 5. Continuous Images of Intervals

Exercises

XIX. The Concept of Dimension

§ 1. 0-Dimensional Spaces

§ 2. Properties of 0-Dimensional Metric Separable Spaces

§ 3. n-Dimensional Spaces

§ 4. Properties of n-Dimensional Metric Separable Spaces

Exercises

XX. Simplexes and Their Properties

§ 1. Simplexes

§ 2. Simplicial Subdivision

§ 3. Dimension of a Simplex

§ 4. The Fixed Point Theorem

Exercises

XXI. Cuttings of the Plane

§ 1. Auxiliary Properties of Polygonal Arcs

§ 2. Cuttings

§ 3. Complex Functions Which Vanish Nowhere. Existence of the Logarithm

§ 4. Auxiliary Theorems

§ 5. Corollaries to the Auxiliary Theorems

§ 6. Theorems on the Cuttings of the Plane

§ 7. Janiszewski Theorems

§ 8. Jordan Theorem

Exercises

Supplement

Elements of Algebraic Topology

Introduction

§ 1. Complexes. Polyhedra. Simplicial Approximation

§ 2. Abelian Groups

§ 3. Categories and Functors

§ 4. Homology Groups of Simplicial Complexes

§ 5. Chain Complexes

§ 6. Homology Groups of Polyhedra

§ 7. Homology Groups with Coefficients

§ 8. Cohomology Groups

Exercises

List of Important Symbols

Index

Other Titles in the Series

## Product details

- No. of pages: 352
- Language: English
- Copyright: © Pergamon 1972
- Published: January 1, 1972
- Imprint: Pergamon
- eBook ISBN: 9781483151632

## About the Editors

### I. S. Sneddon

### M. Stark

## About the Author

### Kazimierz Kuratowski

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