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 Poincaré-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 space.- 9. Completion of subspaces and product spaces.- § 4. Relations between uniform spaces and compact spaces.- 1. Uniformity of compact spaces.- 2. Compactness of uniform spaces.- 3. Compact sets in a uniform space.- 4. Connected sets in a compact space.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Historical Note.- III: Topological Groups.- § 1. Topologies on groups.- 1. Topological groups.- 2. Neighbourhoods of a point in a topological group.- 3. Isomorphisms and local isomorphisms.- § 2. Subgroups, quotient groups, homomorphisms, homogeneous spaces, product groups.- 1. Subgroups of a topological group.- 2. Components of a topological group.- 3. Dense subgroups.- 4. Spaces with operators.- 5. Homogeneous spaces.- 6. Quotient groups.- 7. Subgroups and quotient groups of a quotient group.- 8. Continuous homomorphisms and strict morphisms.- 9. Products of topological groups.- 10. Semi-direct products.- § 3. Uniform structures on groups.- 1. The right and left uniformities on a topological group.- 2. Uniformities on subgroups, quotient groups and product groups.- 3. Complete groups.- 4. Completion of a topological group.- 5. Uniformity and completion of a commutative topological group.- § 4. Groups operating properly on a topological space; compactness in topological groups and spaces with operators.- 1. Groups operating properly on a topological space.- 2. Properties of groups operating properly.- 3. Groups operating freely on a topological space.- 4. Locally compact groups operating properly.- 5. Groups operating continuously on a locally compact space.- 6. Locally compact homogeneous spaces.- § 5. Infinite sums in commutative groups.- 1. Summable families in a commutative group.- 2. Cauchy's criterion.- 3. Partial sums; associativity.- 4. Summable families in a product of groups.- 5. Image of a summable family under a continuous homomorphism.- 6. Series.- 7. Commutatively convergent series.- § 6. Topological groups with operators; topological rings, division rings and fields.- 1. Topological groups with operators.- 2. Topological direct sum of stable subgroups.- 3. Topological rings.- 4. Subrings; ideals; quotient rings; products of rings.- 5. Completion of a topological ring.- 6. Topological modules.- 7. Topological division rings and fields.- 8. Uniformities on a topological division ring.- § 7. Inverse limits of topological groups and rings.- 1. Inverse limits of algebraic structures.- 2. Inverse limits of topological groups and spaces with operators.- 3. Approximation of topological groups.- 4. Application to inverse limits.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- Exercises for § 7.- Historical Note.- IV: Real Numbers.- § 1. Definition of real numbers.- 1. The ordered group of rational numbers.- 2. The rational line.- 3. The real line and real numbers.- 4. Properties of intervals in R.- 5. Length of an interval.- 6. Additive uniformity of R.- § 2. Fundamental topological properties of the real line.- 1. Archimedes' axiom.- 2. Compact subsets of R.- 3. Least upper bound of a subset of R.- 4. Characterization of intervals.- 5. Connected subsets of R.- 6. Homeomorphisms of an interval onto an interval.- § 3. The field of real numbers.- 1. Multiplication in R.- 2. The multiplicative group R*.- 3. nth roots.- § 4. The extended real line.- 1. Homeomorphism of open intervals of R.- 2. The extended line.- 3. Addition and multiplication in R?.- § 5. Real-valued functions.- 1. Real-valued functions.- 2. Real-valued functions defined on a filtered set.- 3. Limits on the right and on the left of a function of a real variable.- 4. Bounds of a real-valued function.- 5. Envelopes of a family of real-valued functions.- 6. Upper limit and lower limit of a real-valued function with respect to a filter.- 7. Algebraic operations on real-valued functions.- § 6. Continuous and semi-continuous real-valued functions.- 1. Continuous real-valued functions.- 2. Semi-continuous functions.- § 7. Infinite sums and products of real numbers.- 1. Families of positive finite numbers summable in R.- 2. Families of finite numbers of arbitrary sign summable in R.- 3. Product of two infinite sums.- 4. Families multipliable in R*.- 5. Summable families and multipliable families in R.- 6. Infinite series and infinite products of real numbers.- § 8. Usual expansions of real numbers; the power of R.- 1. Approximations to a real number.- 2. Expansions of real numbers relative to a base sequence.- 3. Definition of a real number by means of its expansion.- 4. Comparison of expansions.- 5. Expansions to base a.- 6. The power of R.- Exercises for § 1.- Exercises for § 2.- Exercises for § 3.- Exercises for § 4.- Exercises for § 5.- Exercises for § 6.- Exercises for § 7.- Exercises for § 8.- Historical Note.- Index of Notation (Chapters I-IV).- Index of Terminology (Chapters I-IV).
