Index of terminology

The reference numbers indicate the chapter, section and sub-section (or exercise) in that order.

Absolute value of a real number : IV, 1, 6. Absolutely closed space : I, 9, Ex. 18. Absolutely convergent infinite product : IV, 7, 6. Absolutely convergent series : IV, 7, 6. Accessible space : I, 8, Ex. 1. Additive group of the rational line : IV, 1, 2. Additive group of the real line : IV, 1, 3. Additive uniformity of a topological division ring : III, 3, 8. Additive uniformity of the real line : IV, 1, 6. Alexandroff compactification : I, 9, 8. Alexandroff half-line : IV, 2, Exercise 12. Alexandroff’s theorem : I, 9, 8. A-maximal topology : I, 3, Ex. 11. Algebraic complement of a subgroup : III, 6, 2. Alternating series : IV, 7, 6. Approximation to within $\varepsilon$ by defect (excess) : IV, 8, 1. Arbitrarily small subgroups of a topological group : III, 2, Exercise 30. Archimedes’ axiom : IV, 2, 1. Associated bijective continuous homomorphism : III, 2, 8. Associated Hausdorff group : III, 2, 6. Associated Hausdorff module : III, 6, 6. Associated Hausdorff ring : III, 6, 4. Associativity formula : III, 5, 3. Associativity of the sum of a summable family : III, 5, 3. Automorphism, local : III, 1, 3. Automorphism of a topological group : III, 1, 3. Axiom of Archimedes : IV, 2, 1. Axiom of Borel-Lebesgue : I, 9, 1. Axiom of Hausdorff : I, 8, 1.

Base of a filter : I, 6, 3. Base (number) of an expansion of a real number : IV, 8, 5. Base of a topology : I, 1, 3. Base sequence of an expansion of a real number : IV, 8, 2. Bicontinuous mapping : I, 2, 1. Bolzano’s theorem : IV, 6, 1. Borel-Lebesgue axiom : I, 9, 1. Borel-Lebesgue theorem : IV, 2, 2. Bounded above, below (function) : IV, 5, 1. Bounded above, below (set) : IV, 2, 3. Bounded function : IV, 5, 1. Bounded (left, right) subset of a topological ring : III, 6, Exercise 12. Bounded set (in a uniform space) : II, 4, Ex. 7. Bounds, greatest lower and least upper (of a function) : IV, 5, 9.

Canonical mapping of the graph of the equivalence relation defined by a group operating freely : III, 4, 3. Cantor’s theorem : IV, 8, 6. Cantor’s triadic set : IV, 2, 5. Cauchy filter : II, 3, 1. Cauchy filter, minimal : II, 3, 2. Cauchy sequence : II, 3, 1. Cauchy’s condensation test : IV, 7, Exercise 3. Cauchy’s criterion : II, 3, 3. Cauchy’s criterion for series : III, 5, 46. Cauchy’s criterion for summable families : III, 5, 2. Chain (V-) : II, 4, 4. Change of order of summation : III, 5, 3. Close (V-) : II, 1, 1. Closed covering : I, 1, 4. Closed equivalence relation : I, 5, 2. Closed mapping : I, 5, 1. Closed set : I, 1, 4. Closure of a set : I, 1, 6. Cluster point of a filter base : I, 7, 2. Cluster point of a function at a point relative to a subset : I, 7, 5. Cluster point of a function with respect to a directed set : I, 7, 3. Cluster point of a function with respect to a filter : I, 7, 3. Cluster point of a function with respect to a sequence : I, 7, 3. Cluster point of a germ of a function : I, 7, 3. Coarser filter : I, 6, 2. Coarser topology : I, 2, 2. Coarser uniformity : II, 2, 2. Coarsest topology : I, 2, 2.

Commutatively convergent series : III, 5, 7. Compact set : I, 9, 3. Compact space : I, 9, 3. Compactification, Alexandroff or one-point : I, 9, 8. Comparable filters : I, 6, 2. Comparable topologies : I, 2, 2. Comparable uniformities : II, 2, 2. Comparison principle for series : IV, 7, 1. Compatible (division ring structure and topology) : III, 6, 7. Compatible (equivalence relation and group of operators) : III, 2, 4. Compatible (group structure and topology) : III, 1, 1. Compatible (mapping of spaces with operators and homomorphism of groups of operators) : III, 2, 4. Compatible (ordered group structure and topology) : IV, 1, Exercise 1. Compatible (ring structure and topology) : III, 6, 3. Compatible (structure of group with operators and topology) : III, 6, 1. Compatible with a topology : II, 4, 1. Complement, algebraic : III, 6, 2. Complement, topological : III, 6, 2. Complete group : III, 3, 3. Complete ring : III, 6, 5. Complete uniform space : II, 3, 3. Completely Hausdorff (space, topology) : I, 9, Ex. 20. Completely irreducible between two points : IV, 2, Exercise 16. Completion of a Hausdorff topological division ring, field : III, 6, 8. Completion of a Hausdorff topological group : III, 3, 4. Completion of a Hausdorff topological module : III, 6, 6. Completion of a Hausdorff topological ring : III, 6, 5. Completion of a Hausdorff uniform space : II, 3, 7. Component, identity : III, 2, 2. Component of a point : I, 11, 5. Component of a subset : I, 11, 5. Condensation point : I, 9, Ex. 16. Condensation test, Cauchy’s : IV, 7, Exercise 3. Connected component : I, 11, 5. Connected set : I, 11, 1. Connected space : I, 11, 1. Constituents, prime : II, 4, Ex. 19. Contiguous intervals : IV, 2, 5. Continuity, extension by : I, 8, 5. Continuous bijective homomorphism associated with a continuous homomorphism of topological groups : III, 2, 8. Continuous (mapping, function) : I, 2, 1. Continuous section : I, 3, 5.

Continuous with respect to a subspace : I, 3, 2. Continuously, group operating : III, 2, 4. Continuum, power of : IV, 8, 6. Convergent, absolutely : IV, 7, 6. Convergent, commutatively : III, 5, 7. Convergent filter : I, 7, 1. Convergent filter base : I, 7, 1. Convergent infinite product : III, 5, 7. Convergent infinite product of real numbers : IV, 7, 6. Convergent sequence : I, 7, 3. Convergent series : III, 5, 6. Convergent series of real numbers : IV, 7, 6. Coprefilter : I, 6, Ex. 17. Correspondence, proper : I, 10, Ex. 10. Covering, closed : I, 1, 4. Covering, open : I, 1, 1. Criterion, Cauchy’s : II, 3, 3; III, 5, 2 and III, 5, 6. Cube root : IV, 3, 3.

Decimal expansion of a real number : IV, 8, 5. Dense set : I, 1, 6. Direct sum, topological : III, 6, 2. Dirichlet’s function : IV, 6, 2. Discrete division ring : III, 6, 7. Discrete field : III, 6, 7. Discrete group : III, 1, 1. Discrete ring : III, 6, 3. Discrete (topological space, topology) : I, 1, 1. Discrete (uniform space, uniformity) : II, 1, 1. Division ring, topological : III, 6, 7. Dyadic expansion of a real number : IV, 8, 5.

Elementary filter : I, 6, 8. Elementary filter associated with a sequence : I, 6, 8. Elementary set : I, 4, 1. Ends of a locally compact space : I, 11, Ex. 19. Entourage : II, 1, 1. Entourage, symmetric : II, 1, 1. Entourages, fundamental system of : II, 1, 1. Envelope, lower (upper) : IV, 5, 5. Equivalence relation, closed : I, 5, 2. Equivalence relation defined by a group of operators : III, 2, 4. Equivalence relation, Hausdorff : I, 8, 3. Equivalence relation, open : I, 5, 2.

Expansion, decimal : IV, 8, 5. Expansion, dyadic : IV, 8, 5. Expansion, improper : IV, 8, 3. Expansion of a real number with respect to a base sequence : IV, 8, 2. Expansion, terminating : IV, 8, 3. Expansion to base $a$ : IV, 8, 5. Expansion, triadic : IV, 8, 5. Extended real line : IV, 4, 2. Extension of a mapping by continuity : I, 8, 5. Extension of identities, principle of : I, 8, 1. Extension of inequalities, principle of : IV, 5, 2. Exterior of a set : I, 1, 6. Exterior point : I, 1, 6. External semi-direct product : III, 2, 10. External topological semi-direct product : III, 2, 10. Extremally disconnected space : I, 11, Ex. 21.

Factor, general (of an infinite product) : III, 5, 7. Family, locally finite : I, 1, 5. Family, multipliable : III, 5, 1. Family, summable : III, 5, 1. Field, discrete : III, 6, 7. Field of real numbers : IV, 3, 1. Field, $p$-adic : III, 6, Exercise 23. Field, topological : III, 6, 7. Filter : I, 6, 1. Filter base : I, 6, 3. Filter, Cauchy : II, 3, 1. Filter coarser than another : I, 6, 2. Filter comparable with another : I, 6, 2. Filter, convergent : I, 7, 1. Filter, elementary : I, 6, 8. Filter, elementary, associated with a sequence : I, 6, 8. Filter finer than another : I, 6, 2. Filter, Fréchet : I, 6, 1. Filter generated by a set of subsets : I, 6, 2. Filter, induced : I, 6, 5. Filter, intersection : I, 6, 2. Filter, minimal Cauchy : II, 3, 2. Filter, neighbourhood : I, 6, 1. Filter, section : I, 6, 3. Filter strictly coarser than another : I, 6, 2. Filter strictly finer than another : I, 6, 2. Filter subbase : I, 6, 2.

Filtered set : I, 6, 1. Final topology : I, 2, 4. Finer filter : I, 6, 2. Finer topology : I, 2, 2. Finer uniformity : II, 2, 2. Finite open coverings, uniformity of : II, 4, 1. Finite partial sum : III, 5, 1. Finite partitions, uniformity of : II, 2, 2. Finite real number : IV, 4, 2. Finite real-valued function : IV, 5, 1. Freely, group operating : III, 4, 3. Fréchet filter : I, 6, 1. Frontier of a set : I, 1, 6. Frontier point : I, 1, 6. Full subset : I, II, Ex. 14. Function, continuous : I, 2, 1. Function, real-valued (or real) : IV, 5, 1. Function, uniformly continuous : II, 2, 1. Fundamental system of entourages : II, 1, 1. Fundamental system of neighbourhoods : I, 1, 3.

Gelfand ring : III, 6, Exercise 11. General factor of an infinite product : III, 5, 7. General term of a series : III, 5, 6. Geometric progression : IV, 7, 1. Germ of a mapping at a point : I, 6, 10. Germ of a mapping with respect to a filter : I, 6, 9. Germ of a subset at a point : I, 6, 10. Germ of a subset with respect to a filter : I, 6, 9. Greatest lower bound of a real-valued function : IV, 5, 4. Greatest lower bound of a set of topologies : I, 2, 4. Greatest lower bound of a set of uniformities : II, 2, 5. Group, complete : III, 3, 3. Group, discrete : III, 1, 1. Group, Hausdorff, associated with a topological group : III, 2, 6. Group operating freely on a set : III, 4, 3. Group operating trivially on a set : III, 2, 4. Group, paratopological : III, 1, Exercise 4. Group, product topological : III, 2, 9. Group, quasi-topological : III, 2, Exercise 5. Group, quotient topological : III, 2, 6. Group, semi-topological : III, 1, Exercise 2. Group, topological : III, 1, 1. Group, topological, operating continuously on a topological space : III, 2, 4.

Group, topological, operating properly on a topological space : III, 4, 1. Group, topological, with operators : III, 6, 1.

Half-line, Alexandroff : IV, 2, Exercise 12. Hamel base : IV, 6, Exercise 1. Hausdorff completion of a topological group : III, 3, 4. Hausdorff completion of a topological module : III, 6, 6. Hausdorff completion of a topological ring : III, 6, 5. Hausdorff completion of a uniform space : II, 3, 7. Hausdorff equivalence relation: I, 8, 3. Hausdorff group associated with a topological group : III, 2, 6. Hausdorff module associated with a topological module : III, 6, 6. Hausdorff ring associated with a topological ring : III, 6, 4. Hausdorff space : I, 8, 1. Hausdorff topology : I, 8, 1. Hausdorff uniform space associated with a uniform space : II, 3, 8. Hausdorff uniformity : II, 1, 2. Hausdorff’s axiom : I, 8, 1. Homeomorphic topological spaces : I, 1, 1. Homeomorphism : I, 1, 1. Homeomorphism, local : I, 11, Ex. 25. Homogeneous space : III, 2, 5.

Identification, space obtained by : I, 3, 4. Identities, principle of extension of : I, 8, 1. Identity component of a topological group : III, 2, 2. Image, inverse, of a topology : I, 2, 3. Image, inverse, of a uniformity : II, 2, 4. Improper expansion of a real number : IV, 8, 3. Induced by a uniformity (topology) : II, 1, 2. Induced filter : I, 6, 5. Induced topology : I, 2, 3. Induced uniformity : II, 2, 4. Infinite product, absolutely convergent : IV, 7, 6. Infinite product, convergent : III, 5, 7. Infinite product defined by the sequence $(x_n)$ : III, 5, 7. Infinite product of real numbers, convergent : IV, 7, 6. Infinite product whose general factor is $x_n$ : III, 5, 7. Initial topology : I, 2, 3. Initial uniformity : II, 2, 3. Inner automorphism of a topological group : III, 1, 3. Integer, $p$-adic : III, 6, Exercise 23. Integral part of a real number : IV, 8, 2. Interior of a set : I, 1, 6.

Interior point : I, 1, 6. Intersection filter : I, 6, 2. Intersection topology : I, 1, 6. Inverse image of a topology : I, 2, 3. Inverse image of a uniformity : II, 2, 4. Inverse limit of an inverse system of groups (rings) : III, 7, 1. Inverse limit of an inverse system of sets endowed with internal (external) laws : III, 7, 1. Inverse limit of an inverse system of topological groups (modules, rings) : III, 7, 2. Inverse limit of topological spaces : I, 4, 4. Inverse limit of topologies : I, 4, 4. Inverse limit of uniform spaces : II, 2, 7. Inverse limit of uniformities : II, 2, 7. Inverse system of groups (rings) : III, 7, 1. Inverse system of topological groups (rings) : III, 7, 2. Inverse system of topological spaces : I, 4, 4. Inverse system of topologies : I, 4, 4. Inverse system of uniform spaces : II, 2, 7. Inverse system of uniformities : II, 2, 7. Irrational numbers : IV, 1, 2. Irreducible between two points : II, 4, Ex. 19. Isodyne space : I, 2, Ex. 11. Isolated point : I, 1, 6. Isomorphic uniform spaces : II, 1, 1. Isomorphism, local : III, 1, 3. Isomorphism of a topological group onto a topological group : III, 1, 3. Isomorphism of a uniform space onto another : II, 1, 1.

Kolmogoroff space : I, 1, Ex. 2.

Least upper bound of a real-valued function : IV, 5, 4. Least upper bound of a set of topologies : I, 2, 3. Least upper bound of a set of uniformities : II, 2, 5. Left adherent point : IV, 2, Exercise 1. Left bounded : III, 6, Exercise 12. Left inverse of a linear mapping : III, 6, 2. Left topology on an ordered set : I, 1, Ex. 2. Left uniformity of a topological group : III, 3, 1. Length of a bounded interval : IV, 1, 5. Limit, inverse : III, 7, 1 and 2. Limit, lower (upper) : IV, 5, 6. Limit on the left (right) : IV, 5, 3.

Limit (point) of a filter : I, 7, 1. Limit (point) of a filter base : I, 7, 1. Limit (point) of a function at a point : I, 7, 5. Limit (point) of a function relative to a subset : I, 7, 5. Limit (point) of a function with respect to a directed set : I, 7, 3. Limit (point) of a function with respect to a filter : I, 7, 3. Limit (point) of a germ of a function : I, 7, 3. Limit (point) of a sequence : I, 7, 3. Lindelöf space : I, 9, Ex. 14. Line, extended real : IV, 4, 2. Line, rational : I, 1, 2; IV, 1, 2. Line, real : IV, 1, 3. Local automorphism of a topological group : III, 1, 3. Local direct product of topological groups : III, 2, Exercise 26. Local homeomorphism : I, 11, Ex. 25. Local isomorphism of a topological group with another : III, 1, 3. Local product of a family of topological groups, relative to a family of open normal subgroups : III, 2, Exercise 26. Locally bounded topological ring : III, 6, Exercise 12. Locally closed subspace : I, 3, 2. Locally compact space : I, 9, 7. Locally connected space : I, 11, 6. Locally finite family : I, 1, 5. Locally isomorphic topological groups : III, 1, 3. Locally precompact Hausdorff topological group : III, 3, Exercise 8. Locally quasi-compact space : I, 9, Ex. 29. Locally retrobounded topology : III, 6, Exercise 22. Lower envelope of a family of real-valued functions : IV, 5, 5. Lower limit of a real-valued function with respect to a filter : IV, 5, 6. Lower semi-continuous real-valued function : IV, 6, 2. Lower semi-continuous regularization of a real-valued function : IV, 6, 2.

Mapping, bicontinuous : I, 2, 1. Mapping, closed : I, 5, 1. Mapping, continuous : I, 2, 1. Mapping, open : I, 5, 1. Mapping, proper : I, 10, 1. Mapping, uniformly continuous : II, 2, 1. Maximum, relative : IV, 6, 2. Maximum, strict relative : IV, 5, Exercise 10. Minimal Cauchy filter : II, 3, 2. Minimal Hausdorff space : I, 9, Ex. 19. Minimum, relative : IV, 6, 2. Mittag-Leffler’s theorem : II, 3, 5.

Module, topological : III, 6, 6. Morphism of spaces with operators : III, 2, 4. Morphism of topological groups : III, 2, 8. Morphism, strict : III, 2, 8. Monotone limit, theorem of the : IV, 5, 2. Multipliable family of elements of a multiplicative group : III, 5, 1. Multiplicative uniformities of a topological division ring : III, 6, 8.

Neighbourhood filter : I, 6, 1. Neighbourhood of a point : I, 1, 2. Neighbourhood of a set : I, 1, 2. Neighbourhood, symmetric : III, 1, 2. Neighbourhood (V-) : II, 1, 2. Neighbourhoods, fundamental system of : I, 1, 3. Nilpotent, topologically : III, 6, Exercise 9. nth root : IV, 3, 3. Number, finite real : IV, 5, 2. Number, irrational : IV, 1, 3. Number, $p$-adic : III, 6, Exercise 23. Number, real : IV, 1, 3 and (by abuse of language) IV, 4, 2.

One-point compactification : I, 9, 8. Open covering : I, 1, 1. Open equivalence relation : I, 5, 2. Open mapping : I, 5, 1. Open set : I, 1, 1. Opposite of a topological group : III, 1, 1. Orbit space of a space with operators by its group of operators : III, 2, 4. Oscillation of a real-valued function : IV, 6, Exercise 6.

$p$-adic field, integers, numbers, topology, valuation : III, 6, Exercise 23. $p$-adic solenoid : III, 7, Exercise 6. $p$-adic uniformity : II, 1, 1. Paracompact space : I, 9, 10. Paratopological group : III, 1, Exercise 4. Partial sum : III, 5, 3. Pasting together of sets : I, 2, 5. Pasting together of topological spaces : I, 2, 5. Perfect set : I, 1, 6. Periodic expansion of a real number : IV, 8, Exercise 3. Permutable (operations of two groups on a space) : III, 2, 4. Poincaré-Volterra theorem : I, 11, 7. Point : I, 1, 1. Point, cluster : I, 7, 2.

Point, condensation : I, 9, Ex. 16. Point, exterior : I, 1, 6. Point, frontier : I, 1, 6. Point, interior : I, 1, 6. Point, isolated : I, 1, 6. Point, limit : I, 7, 1. Point, singular : II, 4, Ex. 19. Points, V-close : II, 1, 1. Power of the continuum : IV, 8, 6. Precompact set : II, 4, 2. Precompact space : II, 4, 2. Prefilter : I, 6, Ex. 17. Prime constituents : II, 4, Ex. 19. Prime constituents, space of : II, 4, Ex. 19. Prime prefilter : I, 6, Ex. 17. Primitive set : I, 7, Ex. 8. Principle of comparison of series : IV, 7, 1. Principle of extension of identities : I, 8, 1. Principle of extension of inequalities : IV, 5, 2. Product, external semi-direct : III, 2, 10. Product filter : I, 6, 7. Product, local (direct) : III, 2, Exercise 26. Product of a multipliable family : III, 5, 1. Product of topological groups : III, 2, 9. Product of topological groups with operators : III, 6, 1. Product of topological rings : III, 6, 4. Product, semi-direct : III, 2, 10. Product, topological semi-direct : III, 2, 10. Product topological space : I, 4, 1. Product topology : I, 4, 1. Product uniform space : II, 2, 6. Product uniformity : II, 2, 6. Proper correspondence : I, 10, Ex. 10. Proper mapping : I, 10, 1. Properly, group operating : III, 4, 1.

Quasi-compact set : I, 9, 3. Quasi-compact space : I, 9, 1. Quasi-maximal topology : I, 2, Ex. 6. Quasi-ring : III, 6, Exercise 20. Quasi-topological group : III, 2, Exercise 5. Quotient space : I, 3, 4. Quotient space of a space with operators by its group of operators : III, 2, 4. Quotient topological group : III, 2, 6.

Quotient topological group with operators : III, 6, 1. Quotient topological ring : III, 6, 4. Quotient topology : I, 3, 4. Quotient topology of the topology of a group by a subgroup : III, 2, 5. Quotient topology of the topology of a space with operators by its group of operators : III, 2, 4.

Radical of a commutative topological group : III, 2, Exercise 28. Radical group : III, 2, Exercise 28. Rational line : I, 1, 2; IV, 1, 2. Rational number space of $n$ dimensions : I, 4, 1. Real function : IV, 5, 1. Real line : IV, 1, 3. Real line, extended : IV, 4, 2. Real number : IV, 1, 3 and (by abuse of language) IV, 4, 2. Real-valued function : IV, 5, 1. Real-valued function bounded above (below) : IV, 5, 4. Real-valued function, finite : IV, 5, 1. Real-valued function, lower (upper) semi-continuous : IV, 6, 2. Regular open set : I, 8, Ex. 20. Regular space : I, 8, 4. Regular topology : I, 8, 4. Regularization, lower (upper) semi-continuous : IV, 6, 2. Relative maximum (minimum) : IV, 6, 2. Relatively compact : I, 9, 3. Relatively quasi-compact : I, 9, 3. Residual closed subgroup : III, 2, Exercise 28. Residue of a series : III, 5, 6. Restricted associativity of series : III, 5, 7. Retrobounded set : III, 6, Exercise 22. Right bounded : III, 6, Exercise 12. Right inverse of a linear mapping : III, 6, 2. Right topology (on an ordered set) : I, 1, Ex. 2. Right uniformity of a topological group : III, 3, 1. Ring, complete : III, 6, 5. Ring, discrete : III, 6, 3. Ring, Gelfand : III, 6, Exercise 11. Ring, Hausdorff, associated with a topological ring : III, 6, 4. Ring, locally bounded topological : III, 6, Exercise 12. Ring, product topological : III, 6, 4. Ring, quotient topological : III, 6, 4. Ring, topological : III, 6, 3. Ring, topological, Hausdorff completion of : III, 6, 5. Roots (square, cube, $nth$) : IV, 3, 3.

σ-compact : I, 9, 9. Saturation of a subset with respect to a group to a group of operators : III, 2, 4. Section, continuous : I, 3, 5. Section filter : I, 6, 3. Semi-continuous, lower (upper) : IV, 6, 2. Semi-direct product of subgroups : III, 2, 10. Semi-topological goup : III, 1, Exercise 2. Semi-regular (space, topology) : I, 8, Ex. 20. Sequence, Cauchy : II, 3, 1. Sequence convergent : I, 7, 3. Series, absolutely convergent : IV, 7, 6. Series, alternating : IV, 7, 6. Series, commutatively convergent : III, 5, 7. Series, convergent : III, 5, 6 and IV, 7, 6. Series defined by the sequence $(x_n)$ : III, 5, 6. Series whose general term is $x_n$ : III, 5, 6. Set, bounded : II, 4, Ex. 7. Set bounded above (below) : IV, 2, 3. Set, Cantor’s triadic : IV, 2, 5. Set, closed : II, 1, 4. Set, compact : I, 9, 3. Set, connected : I, 11, 1. Set, dense : I, 1, 6. Set, elementary : I, 4, 1. Set, filtered : I, 6, 1. Set, locally closed : I, 3, 3. Set, open : I, 1, 1. Set, perfect : I, 1, 6. Set, precompact : II, 4, 2. Set, quasi-compact : I, 9, 3. Set, regular open : I, 8, Ex. 20. Set, relatively compact : I, 9, 3. Set, relatively quasi-compact : I, 9, 3. Set, totally disconnected : I, 11, 5. Set underlying a topological space : I, 1, 1. Sign of a real number : IV, 3, 2. Singular point : II, 4, Ex. 19. Small (V-) : II, 3, 1. Solenoid, $p$-adic : III, 7, Exercise 6. Solvable space : I, 2, Ex. 11. Space, absolutely closed : I, 9, Ex. 18. Space, accessible : I, 8, Ex. 1. Space, compact : I, 9, 1.

Space, complete : II, 3, 3. Space, completely Hausdorff : I, 9, Ex. 20. Space, connected : I, 11, 1. Space, extremally disconnected : I, 11, Ex. 21. Space, Hausdorff : I, 8, 1. Space, Hausdorff, associated with a uniform space : II, 3, 8. Space, homogeneous : III, 2, 5. Space, isodyne : I, 2, Ex. 11. Space, Kolmogoroff : I, 1, Ex. 2. Space, Lindelöf : I, 9, Ex. 14. Space, locally compact : I, 9, 7. Space, locally compact σ-compact : I, 9, 9. Space, locally connected : I, 11, 6. Space, locally quasi-compact : I, 9, Ex. 29. Space, minimal Hausdorff : I, 9, Ex. 19. Space, n-dimensional rational : I, 4, 1. Space of orbits of a group operating continuously on a topological space : III, 2, 4. Space of prime constituents : II, 4, Ex. 19. Space, paracompact : I, 9, 10. Space, precompact : II, 4, 2. Space, product topological : I, 4, 1. Space, product uniform : II, 2, 6. Space, quasi-compact : I, 9, 1. Space, quotient : I, 3, 4; III, 2, 4. Space, rational n-dimensional : I, 4, 1. Space, regular : I, 8, 4. Space, semi-regular : I, 8, Ex. 20. Space, solvable : I, 2, Ex. 11. Space, submaximal : I, 8, Ex. 22. Space, sum : I, 2, 4. Space, topological : I, 1, 1. Space, topological homogeneous : III, 2, 5. Space, topological, underlying a uniform space : II, 1, 2. Space, totally disconnected : I, 11, 5. Space, ultrafilter : I, 9, Ex. 26. Space, ultraregular : I, 8, Ex. 25. Space, uniform : II, 1, 1. Space, uniformizable : II, 4, 1. Square root : IV, 3, 3. Stabilizer : III, 2, 4. Strict morphism : III, 2, 8. Strict relative maximum : IV, 5, Exercise 10. Strictly coarser filter : I, 6, 2.

Strictly coarser topology : I, 2, 2. Strictly coarser uniformity : II, 2, 2. Strictly finer filter : I, 6, 2. Strictly finer topology : I, 2, 2. Strictly finer uniformity : II, 2, 2. Subbase of a filter : I, 6, 2. Subbase of a topology : I, 2, 3. Submaximal (space, topology) : I, 8, Ex. 22. Subspace (of a topological space) : I, 3, 1. Subspace, locally closed : I, 3, 3. Subspace of a uniform space : II, 2, 4. Sum, finite partial : III, 5, 1. Sum of a family of points of a commutative topological group : III, 5, 1. Sum of a series : III, 5, 6. Sum, partial : III, 5, 3. Sum (topological space, topology) : I, 2, 4. Summable family : III, 5, 1. Symmetric entourage : II, 1, 1. Symmetric neighbourhood : III, 1, 2. Symmetry : III, 1, 1.

Terminating expansion (of a real number) : IV, 8, 3. Theorem, Alexandroff’s : I, 9, 8. Theorem, Mittag-Leffler’s : II, 3, 5. Theorem, Poincaré-Volterra : I, 11, 7. Theorem, Tychonoff’s : I, 9, 5. Theorem of Bolzano : IV, 6, 1. Theorem of Borel-Lebesgue : IV, 2, 2. Theorem of Cantor : IV, 8, 6. Theorem of the monotone limit : IV, 5, 2. Theorem of Weierstrass : IV, 6, 1. Topological complement of a stable subgroup : III, 6, 2. Topological direct sum of stable subgroups : III, 6, 2. Topological division ring : III, 6, 7. Topological field : III, 6, 7. Topological group : III, 1, 1. Topological group with operators : III, 6, 1. Topological homogeneous space : III, 2, 5. Topological module : III, 6, 6. Topological ring : III, 6, 3. Topological semi-direct product : III, 2, 10. Topological space : I, 1, 1. Topological structure : see Topology. Topologically nilpotent (element, ideal) : III, 6, Exercise 9.

Topology : I, 1, 1. Topology, A-maximal : I, 3, Ex. 11. Topology associated with a filter : I, 6, 5. Topology coarser than another : I, 2, 2. Topology, coarsest : I, 2, 2. Topology comparable with another : I, 2, 2. Topology, completely Hausdorff : I, 9, Ex. 20. Topology, discrete : I, 1, 1. Topology, final : I, 2, 4. Topology finer than another : I, 2, 2. Topology generated by a set of subsets : I, 2, 3. Topology, Hausdorff : I, 8, 1. Topology, induced : I, 2, 3. Topology induced by a uniformity : II, 1, 2. Topology, initial : I, 2, 3. Topology, intersection : I, 1, 6. Topology, left (on an ordered set) : I, 1, Ex. 2. Topology, locally bounded : III, 6, Exercise 12. Topology, locally retrobounded : III, 6, Exercise 22. Topology, $p$-adic : III, 6, Exercise 23. Topology, product : I, 2, 3. Topology, quasi-maximal : I, 2, Ex. 6. Topology, quotient : I, 2, 4. Topology, regular : I, 8, 4. Topology, right (on an ordered set) : I, 1, Ex. 2. Topology, semi-regular, associated with a topology : I, 8, Ex. 20. Topology strictly coarser than another : I, 2, 2. Topology strictly finer than another : I, 2, 2. Topology, submaximal : I, 8, Ex. 22. Topology, sum : I, 2, 4. Topology, ultraregular : I, 8, Ex. 25. Topology, uniformizable : II, 4, 1. Totally adherent : I, 9, Ex. 17. Totally disconnected (set, space) : I, 11, 5. Triadic expansion of a real number : IV, 8, 5. Triadic set, Cantor’s : IV, 2, 6. Triangle inequality : IV, 1, 1. Trivial ultrafilter : I, 6, 4. Trivially, group operating : III, 2, 4. Two-sided uniformity on a topological group : III, 2, Exercise 6. Tychonoff’s theorem : I, 9, 5.

Ultrafilter : I, 6, 4. Ultrafilter space : I, 9, Ex. 26.

Ultrafilter, trivial : I, 6, 4. Ultraregular (space, topology) : I, 8, Ex. 25. Underlying set of a topological space : I, 1, 1. Underlying topological space of a uniform space : II, 1, 2. Uniform structure : see Uniformity. Uniformities, multiplicative (of a topological division ring) : III, 6, 8. Uniformity : II, 1, 1. Uniformity, additive (of a topological division ring) : III, 6, 8. Uniformity, additive (of the real line) : IV, 1, 6. Uniformity coarser than another : II, 2, 2. Uniformity comparable with another : II, 2, 2. Uniformity compatible with a topology : II, 4, 1. Uniformity, discrete : II, 1, 1. Uniformity finer than another : II, 2, 2. Uniformity, Hausdorff : II, 1, 2. Uniformity, induced : II, 2, 4. Uniformity, left (right) (of a topological group) : III, 3, 1. Uniformity of finite open coverings : II, 4, 1. Uniformity of finite partitions : II, 2, 2. Uniformity, $p$-adic : II, 1, 1. Uniformity, product : II, 2, 6. Uniformity strictly coarser than another : II, 2, 2. Uniformity strictly finer than another : II, 2, 2. Uniformity, two-sided : III, 3, Exercise 6. Uniformizable (topological space, topology) : II, 4, 1. Uniformly bounded above (below) (family of real-valued functions) : IV, 5, 5. Uniformly continuous mapping : II, 2, 1. Upper envelope of a family of real-valued functions : IV, 5, 5. Upper limit of a real-valued function with respect to a filter : IV, 5, 5. Upper semi-continuous real-valued function : IV, 6, 2. Upper semi-continuous regularization of a real-valued function : IV, 6, 2.

Valuation, $p$-adic : III, 6, Exercise 23. Value, absolute : IV, 1, 6. Value of a germ of a mapping at a point : I, 6, 10. V-chain : II, 4, 4. V-close (points) : II, 1, 1. V-neighbourhood of a set : II, 1, 2. V-small : II, 3, 1.

Weierstrass's theorem : IV, 6, 1.