
Preface to the first edition 
vii 

Preface to the revised edition 
viii 




Introduction 

I.1 
Algebra of sets. Functions 
1 
I.2 
Cardinal numbers 
3 
I.3 
Order relations. Ordinal numbers 
4 
I.4 
The axiom of choice 
8 
I.5 
Real numbers 
10 




Chapter 1: Topological spaces 

1.1 
Topological spaces. Open and closed sets. Bases. Closure and interior of a set 
11 
1.2 
Methods of generating topologies 
20 
1.3 
Boundary of a set and derived set. Dense and nowhere dense sets. Borel sets 
24 
1.4 
Continuous mappings. Closed and open mappings. Homeomorphisms 
27 
1.5 
Axioms of separation 
36 
1.6 
Convergence in topological spaces: Nets and filters. Sequential
and Fréchet spaces 
49 
1.7 
Problems 
56 




Chapter 2: Operations on topological spaces 

2.1 
Subspaces 
65 
2.2 
Sums 
74 
2.3 
Cartesian products 
77 
2.4 
Quotient spaces and quotient mappings 
90 
2.5 
Limits of inverse systems 
98 
2.6 
Function spaces I: The
topology of uniform convergence on R^{X} and the topology of pointwise convergence 
105 
2.7 
Problems 
112 




Chapter 3: Compact spaces 

3.1 
Compact spaces 
123 
3.2 
Operations on compact spaces 
136 
3.3 
Locally compact spaces
and kspaces 
148 
3.4 
Function spaces II: The compactopen topology 
156 
3.5 
Compactifications 
166 
3.6 
The CechStone compactification and the Wallman extension 
172 
3.7 
Perfect mappings 
182 
3.8 
Lindelöf spaces 
192 
3.9 
Cechcomplete spaces 
196 
3.10 
Countably compact spaces, pseudocompact spaces and sequentially compact spaces 
202 
3.11 
Realcompact spaces 
214 
3.12 
Problems 
220 




Chapter 4: Metric and metrizable spaces 

4.1 
Metric and metrizable spaces 
248 
4.2 
Operations on metrizable spaces 
258 
4.3 
Totally bounded and complete metric spaces. Compactness in metric spaces 
266 
4.4 
Metrization theorems I 
280 
4.5 
Problems 
288 




Chapter 5: Paracompact spaces 

5.1 
Paracompact spaces 
299 
5.2 
Countably paracompact spaces 
316 
5.3 
Weakly and strongly paracompact spaces 
322 
5.4 
Metrization theorems II 
329 
5.5 
Problems 
337 




Chapter 6: Connected spaces 

6.1 
Connected spaces 
352 
6.2 
Various kinds of disconnectedness 
360 
6.3 
Problems 
372 




Chapter 7: Dimension of topological spaces 

7.1 
Definitions and basic properties of dimensions ind, Ind, and dim 
383 
7.2 
Further properties of the dimension dim 
394 
7.3 
Dimension of metrizable spaces 
402 
7.4 
Problems 
418 




Chapter 8: Uniform spaces and proximity spaces 

8.1 
Uniformities and uniform spaces 
426 
8.2 
Operations on uniform spaces 
438 
8.3 
Totally bounded and complete uniform spaces. Compactness in uniform spaces 
444 
8.4 
Proximities and proximity spaces 
451 
8.5 
Problems 
460 




Bibliography 
469 




Tables 


Relations between main classes of topological spaces 
508 

Invariants of operations 
509 

Invariants and inverse invariants of mappings 
510 




List of special symbols 
511 

Author index 
514 

Subject index 
520 