1 Introduction

1.1 Explanatory Notes

1.2 n-Space

1.3 Abstraction

2 Topological Space

2.1 Topological space

2.2 Semimetric and metric space

2.3 Semimetric and metric topologies

2.4 Natural topologies and metrics

2.5 Notation and terminology

2.6 Base and subbase

3 Convergence

3.1 Sequences

3.2 Filters

3.3 Partially ordered sets

3.4 Nets

3.5 Arithmetic of nets

4 Separation Axioms

4.1 Separation by open sets

4.2 Continuity

4.3 Separation by continuous functions

5 Topological Concepts

5.1 Topological properties

5.2 Connectedness

5.3 Separability

5.4 Compactness

6 Sup, Weak, Product and Quotient Topologies

6.1 Introduction

6.2 Sup topologies

6.3 Weak topologies

6.4 Products

6.5 Quotients

6.6 Continuity

6.7 Separation

7 Compactness

7.1 Countable and sequential compactness

7.2 Compactness in semimetric space

7.3 Ultrafilters

7.4 Products

8 Compactification

8.1 The one-point compactification

8.2 Embeddings

8.3 The Stone-Cech compactification

8.4 Compactifications

8.5 C- and C*-embedding

8.6 Realcompact spaces

9 Complete Semimetric Space

9.1 Completeness

9.2 Completion

9.3 Baire category

10 Metrization

10.1 Separable spaces

10.2 Local finiteness

10.3 Metrization

11 Uniformity

11.1 Uniform space

Il.2 Uniform continuity

11.3 Uniform concepts

11.4 Uniformization

11.5 Metrization and completion

12 Topological Groups

12.1 Group topologies

12.2 Group concepts

12.3 Quotients

12.4 Topological vector spaces

13 Function Spaces

13.1 The compact open topology

13.2 Topologies of uniform convergence

13.3 Equicontinuity

13.4 Weak compactness

14 Miscellaneous Topics

14.1 Extremally disconnected spaces

14.2 The Gleason map

14.3 Categorical algebra

14.4 Paracompact spaces

14.5 Ordinal spaces

14.6 The Tychonoff plank

14.7 Completely regular and normal spaces

Appendix: Tables of Theorems and Counterexamples

Bibliography

Index