ContentsPreface to the first editionPreface to the revised editionIntroductionI.1. Algebra of sets. FunctionsI.2. Cardinal numbersI.3 Order relations. Ordinal numbersI.4. The axiom of choiceI.5. Real numbersHistorical and bibliographical notes

Chapter 1. Topological spaces1.1. Topological spaces. Open and closed sets. Bases. Closure and interior of a setHistorical and bibliographical notesExercises

1.2. Methods of generating topologiesHistorical and bibliographical notesExercises

1.3. Boundary of a set and derived set. Dense and nowhere dense sets. Borel setsHistorical and bibliographical notesExercises

1.4. Continuous mappings. Closed and open mappings. Homeomorphisms. Historical and bibligraphic notes Exercises

1.5. Axioms of separation. Historical and bibligraphic notes Exercises

1.6. Convergence in topological spaces: nets and filters. Sequential and Frechet spaces. Historical and bibligraphic notes. Exercises.

1.7. Problems.Left and right topology on an ordered set.Linearly ordered spaces I.Borel sets I.Normally palced sets I.Urysohn spaces and semiregular spaces.The Cantor-Benedixson theorem.Cardinal functions I.Semicontinuous functions I.Set-valued mappings I.Topologies described by sequences.

Chapter 2. Operations on Topological Spaces.2.1. Subspaces. Historical and bibligraphic notes. Exercises.

2.2. Sums. Historical and bibligraphic notes. Exercises.

2.3. Cartesian products. Historical and bibligraphic notes. Exercises.

2.4. Quotient spaces and quotient mappings. Historical and bibligraphic notes. Exercises.

2.5. Limits of inverse systems. Historical and bibligraphic notes. Exercises.

2.6. Function spaces I: the topology of uniform convergence on R^X and the topology of pointwise convergence. Historical and bibligraphic notes. Exercises.

2.7. ProblemsLocally closed setsSeparated F-sigma-sets in normal spaces. Normally placed sets II.Semicontinuous functions II.Linearly ordered spaces II.Urysohn spaces and semiregular spaces II.Embedding in cartesian products.Cardinal functions II.Functions on cartesian products.Sigma-products I.A regular space on which every continuous real-valued function is constant.Inverse systems I.Spaces of closed subsets I.Set-valued mappings II.

Chapter 3. Compact spaces.3.1. Compact spaces. Historical and bibligraphic notes. Exercises.

3.2. Operations on compact spaces. Historical and bibligraphic notes. Exercises.

3.3. Locally compact spaces and k-spaces. Historical and bibligraphic notes. Exercises.

3.4. Function spaces II: the compact-open topology. Historical and bibligraphic notes. Exercises.

3.5. Compactifications. Historical and bibligraphic notes. Exercises.

3.6. The Chech-Stone compactification and the Wallman extension. Historical and bibligraphic notes. Exercises.

3.7. Perfect mappings. Historical and bibligraphic notes. Exercises.

3.8. Lindelof spaces. Historical and bibligraphic notes. Exercises.

3.9. Chech-complete spaces. Historical and bibligraphic notes. Exercises.

3.10. Countably compact spaces, pseudocompact spaces and sequentially compact spaces. Historical and bibligraphic notes.Exercises.

3.11. Realcompact spaces. Historical and bibligraphic notes. Exercises.

3.12. Problems. Further characterizations of compactness: complete accumulation points and Alexander subbase theoremLinearly ordered spaces III.H-closed and H-minimal spaces.Cardinal functions III.Dyadic spaces I.Inverse systems II.Around the Kuratowski and the Whitehead theoremsNormality and related properties in Cartesian products II.Compactifications.Parovichenko spaces. The long line and the long segment.The Tychonoff plank and related spaces. The Chech-Stone compactification of Cartesian products.Rings of continuous functions and compactifications.Countable compactness.Sigma-products II.Normally placed sets III.Regularly placed setsSpaces of closed subsets II.Set-valued mappings III.

Chapter 4. Metric and metrizable spaces. 4.1. Metric and metrizable spaces. Historical and bibligraphic notes. Exercises.

4.2. Operations on metrizable spaces. Historical and bibligraphic notes. Exercises.

4.3. Totally bounded and complete metric spaces. Compactness in metric spaces. Historical and bibligraphic notes. Exercises.

4.4. Metrization theorems I. Historical and bibligraphic notes. Exercises.

4.5 Problems. Extending closed and open sets.R^n is homogeneous with respect to countable dense subsets.The topology of pointwise convergence and metrics.Expanding and contracting mappings of metric spaces.Every dense-in-itself completely metrizable space contains the Cantor set.A direct construction of the completion.Borel sets II.Dyadic spaces IISigma-products III.Extending closed and open mappings.Normality and related properties in Cartesian products III. Invariance of metrizability under open and quotient mappings.Closed images of metrizable spaces.Extending functions and metrics.The space R^I is metrically universal for all separable metric spacesSpaces of closed subsets III.

Chapter 5. Paracompact spaces.5.1. Paracompact spaces. Historical and bibliographic notes. Exercises.

5.2. Countably paracompact spaces. Historical and bibliographic notes. Exercises.

5.3. Weakly and strongly paracompact spaces. Historical and bibliographic notes. Exercises.

5.4. Metrization theorems II. Historical and bibliographic notes. Exercises.

5.5. Problems. Collectionwise normality.Operation X_MAround Bing's and Michael's examples.Paracompactness of Cartesian products.Paracompact spaces with G_\delta-diagonals.Paracompactness and Chech-complete spaces.Paracompactness and realcompact spaces.Compact-covering mappingsIrreducible mappingsParacompactness of function spaces.F_\sigma-sets in countably paracompact spaces.Countable paracompactness in normal spaces.Extending locally finite families of sets.Normality and related properties in Cartesian products IV.Semicontinuous functions III.The Borsuk homotopy extension theorem.Linearly ordered spaces IV.Every pseudocompact weakly paracompact space is compact.

Chapter 6. Connected spaces.6.1. Connected spaces. Historical and bibliographic notes. Exercises.

6.2. Various kinds of disconnectedness. Historical and bibliographic notes. Exercises.

6.3. Problems. A characterization of connectedness.Linearly ordered spaces V.Local connectedness.A topological characterization of the closed interval.Pathwise connectedness and local pathwise connectedness.Quasi-components of Cartesian products.Inverse systems III.Urysohn spaces and semiregular spaces III.Extremal disconnectedness and axioms of separationProjective spaces and projective resolutions (absolutes)Extremal disconnectedness and Cartesian products.Spaces of closed subsets IV.The Knaster-Kuratowski fan.Around Erdos' example.An inverse sequence of strongly zero-dimensional spaces whose limit is not strongly zero-dimensional.

Chapter 7. Dimension of topological spaces.7.1. Definitions and basic properties of dimensions ind, Ind and dim. Historical and bibliographic notes. Exercises.

7.2. Further properties of the dimension dim. Historical and bibliographic notes. Exercises.

7.3. Dimension of metrizable spaces. Historical and bibliographic notes. Exercises.

Appendix: proof of the Brouwer fixed-point theorem.7.4. Problems. The addition theorems.A Tychonoff space Z and a closed subspace M of Z such that dim Z