# Book:Ryszard Engelking/General Topology/Revised and Completed Edition

## Ryszard Engelking: *General Topology (Revised and Completed Edition)*

Published $1989$, **Heldermann**

- ISBN 3-88538-006-4.

### Subject Matter

### Contents

- Preface to the first edition
- Preface to the revised edition

**Introduction**

- I.1 Algebra of sets. Functions
- I.2 Cardinal numbers
- I.3 Order relations. Ordinal numbers
- I.4 The axiom of choice
- I.5 Real numbers

**Chapter 1: Topological spaces**

- 1.1 Topological spaces. Open and closed sets. Bases. Closure and interior of a set
- 1.2 Methods of generating topologies
- 1.3 Boundary of a set and derived set. Dense and nowhere dense sets. Borel sets
- 1.4 Continuous mappings. Closed and open mappings. Homeomorphisms
- 1.5 Axioms of separation
- 1.6 Convergence in topological spaces: Nets and filters. Sequential and Fréchet spaces
- 1.7 Problems

**Chapter 2: Operations on topological spaces**

- 2.1 Subspaces
- 2.2 Sums
- 2.3 Cartesian products
- 2.4 Quotient spaces and quotient mappings
- 2.5 Limits of inverse systems
- 2.6 Function spaces I: The topology of uniform convergence on RX and the topology of pointwise convergence
- 2.7 Problems

**Chapter 3: Compact spaces**

- 3.1 Compact spaces
- 3.2 Operations on compact spaces
- 3.3 Locally compact spaces and $k$-spaces
- 3.4 Function spaces II: The compact-open topology
- 3.5 Compactifications
- 3.6 The Cech-Stone compactification and the Wallman extension
- 3.7 Perfect mappings
- 3.8 Lindelöf spaces
- 3.9 Cech-complete spaces
- 3.10 Countably compact spaces, pseudocompact spaces and sequentially compact spaces
- 3.11 Realcompact spaces
- 3.12 Problems

**Chapter 4: Metric and metrizable spaces**

- 4.1 Metric and metrizable spaces
- 4.2 Operations on metrizable spaces
- 4.3 Totally bounded and complete metric spaces. Compactness in metric spaces
- 4.4 Metrization theorems I
- 4.5 Problems

**Chapter 5: Paracompact spaces**

- 5.1 Paracompact spaces
- 5.2 Countably paracompact spaces
- 5.3 Weakly and strongly paracompact spaces
- 5.4 Metrization theorems II
- 5.5 Problems

**Chapter 6: Connected spaces**

- 6.1 Connected spaces
- 6.2 Various kinds of disconnectedness
- 6.3 Problems

**Chapter 7: Dimension of topological spaces**

- 7.1 Definitions and basic properties of dimensions ind, Ind, and dim
- 7.2 Further properties of the dimension dim
- 7.3 Dimension of metrizable spaces
- 7.4 Problems

**Chapter 8: Uniform spaces and proximity spaces**

- 8.1 Uniformities and uniform spaces
- 8.2 Operations on uniform spaces
- 8.3 Totally bounded and complete uniform spaces. Compactness in uniform spaces
- 8.4 Proximities and proximity spaces
- 8.5 Problems

**Bibliography**

**Tables**- Relations between main classes of topological spaces
- Invariants of operations
- Invariants and inverse invariants of mappings

- List of special symbols
- Author index
- Subject index