General Topology: Chapters 1-4 (Paperback, 1995)

'of the Elements of Mathematics Series.- I. Topological Structures.- 1. Open sets, neighbourhoods, closed sets.- 1. Open sets.- 2. Neighbourhoods.- 3. Fundamental systems of neighbourhoods; bases of a topology.- 4. Closed sets.- 5. Locally finite families.- 6. Interior, closure, frontier of a set; dense sets.- 2. Continuous functions.- 1. Continuous functions.- 2. Comparison of topologies.- 3. Initial topologies.- 4. Final topologies.- 5. Pasting together of topological spaces.- 3. Subspaces, quotient spaces.- 1. Subspaces of a topological space.- 2. Continuity with respect to a subspace.- 3. Locally closed subspaces.- 4. Quotient spaces.- 5. Canonical decomposition of a continuous mapping.- 6. Quotient space of a subspace.- 4. Product of topological spaces.- 1. Product spaces.- 2. Section of an open set; section of a closed set, projection of an open set. Partial continuity.- 3. Closure in a product.- 4. Inverse limits of topological spaces.- 5. Open mappings and closed mappings.- 1. Open mappings and closed mappings.- 2. Open equivalence relations and closed equivalence relations.- 3. Properties peculiar to open mappings.- 4. Properties peculiar to closed mappings.- 6. Filters.- 1. Definition of a filter.- 2. Comparison of filters.- 3. Bases of a filter.- 4. Ultrafilters.- 5. Induced filter.- 6. Direct image and inverse image of a filter base.- 7. Product of filters.- 8. Elementary filters.- 9. Germs with respect to a filter.- 10. Germs at a point.- 7. Limits.- 1. Limit of a filter.- 2. Cluster point of a filter base.- 3. Limit point and cluster point of a function.- 4. Limits and continuity.- 5. Limits relative to a subspace.- 6. Limits in product spaces and quotient spaces.- 8. Hausdorff spaces and regular spaces.- 1. Hausdorff spaces.- 2. Subspaces and products of Hausdorff spaces.- 3. Hausdorff quotient spaces.- 4. Regular spaces.- 5. Extension by continuity; double limit.- 6. Equivalence relations on a regular space.- 9. Compact spaces and locally compact spaces.- 1. Quasi-compact spaces and compact spaces.- 2. Regularity of a compact space.- 3. Quasi-compact sets; compact sets; relatively compact sets.- 4. Image of a compact space under a continuous mapping.- 5. Product of compact spaces.- 6. Inverse limits of compact spaces.- 7. Locally compact spaces.- 8. Embedding of a locally compact space in a compact space.- 9. Locally compact ?-compact spaces.- 10. Paracompact spaces.- 10. Proper mappings.- 1. Proper mappings.- 2. Characterization of proper mappings by compactness properties.- 3. Proper mappings into locally compact spaces.- 4. Quotient spaces of compact spaces and locally compact spaces.- 11. Connectedness.- 1. Connected spaces and connected sets.- 2. Image of a connected set under a continuous mapping.- 3. Quotient spaces of a connected space.- 4. Product of connected spaces.- 5. Components.- 6. Locally connected spaces.- 7. Application : the Poincare-Vol terra theorem.- Exercises for 1.- Exercises for 2.- Exercises for 3.- Exercises for 4.- Exercises for 5.- Exercises for 6.- Exercises for 7.- Exercises for 8.- Exercises for 9.- Exercises for 10.- Exercises for 11.- Historical Note.- II. Uniform Structures.- 1. Uniform spaces.- 1. Definition of a uniform structure.- 2. Topology of a uniform space.- 2. Uniformly continuous functions.- 1. Uniformly continuous functions.- 2. Comparison of uniformities.- 3. Initial uniformities.- 4. Inverse image of a uniformity; uniform subspaces.- 5. Least upper bound of a set of uniformities.- 6. Product of uniform spaces.- 7. Inverse limits of uniform spaces.- 3. Complete spaces.- 1. Cauchy filters.- 2. Minimal Cauchy filters.- 3. Complete spaces.- 4. Subspaces of complete spaces.- 5. Products and inverse limits of complete spaces.- 6. Extension of uniformly continuous functions.- 7. The completion of a uniform space.- 8. The Hausdorff uniform space associated with a uniform spa