Index of terminology

The reference numbers indicate the chapter, section and subsection or Exercise, in that order.

Abscissa : VI, 1, 4. Abscissas, axis of : VI, 1, 4. Absolute value : IX, 3, 2. Absolute value, improper : IX, 3, 2. Absolute value, $p$-adic : IX, 3, 2. Absolute values, equivalent : IX, 3, 2. Absolute value of a complex number : VIII, 1, 1. Absolutely convergent infinite product of complex numbers : VIII, 3, 3. Absolutely convergent series (in a normed space) : IX, 3, 6. Absolutely convergent series of points in $\mathbf{R}^n$ : VII, 3, 2. Absolutely summable family (in a normed space) : IX, 3, 6. Acute angular sector : VIII, 2, 5. Additive group of $\mathbf{R}^n$ : VI, 1, 2. Additive group of real numbers mod $a$ : V, 1, 2. Additive uniformity of $\mathbf{R}^n$ : VII, 1, 2. Admissible subset : IX, 6, Exercise 12. Affine linear varieties, orthogonal : VI, 2, 2. Algebra, normed : IX, 3, 7. Algebraic norm of a complex number : VIII, 1, 1. Algebraic number : VIII, 1, Exercise 2. Almost-open mapping : IX, 5, Exercise 24. Almost-open set : IX, 5, Exercise 6. Almost-periodic function : X, 3, Exercise 28. Almost-periodic (point) : X, 3, Exercise 27. Amplitude of a complex number : VIII, 2, 2 and VIII, 2, 3. Angle, flat : VIII, 2, 2. Angle of a pair of rays : VIII, 2, 2 and VIII, 2, 3. Angle of an angular sector : VIII, 2, 5. Angle, positive right : VIII, 2, 2.

Angular sector (acute, right, obtuse, flat, salient, re-entrant): VIII, 2, 5. Approximation, uniform: X, 4, 1. Archimedean (linearly ordered group): V, 3, Exercise 1. Archimedes’ axiom: V, 2. Argument of a complex number: VIII, 2, 2. Ascoli’s theorem: X, 2, 5. Associated subgroup of a subgroup of $\mathbf{R}^n$: VII, 1, 3. Axiom of Archimedes: V, 2. Axis, coordinate: VI, 1, 4. Axis, imaginary: VIII, 1, 2. Axis of abscissas: VI, 1, 4. Axis of ordinates: VI, 1, 4. Axis, real: VIII, 1, 2.

Baire space: IX, 5, 3. Baire’s theorem: IX, 5, 3. Ball (closed, open): IX, 2, 2. Ball, Euclidean (open, closed): VI, 2, 3. Ball, unit: VI, 2, 3, and IX, 3, 3. Base of a system of angular measure: VIII, 2, 3. Base of a system of logarithms: V, 4, 1. Basis, canonical (of $\mathbf{R}^n$): VI, 1, 3. Bisector of an angular sector: VIII, 2, 5. Borel mapping: IX, 6, Exercise 16. Borel mapping of class $\alpha$: IX, 6, Exercise 16. Borel set: IX, 6, 3. Bounded convergence, uniformity of: X, 1, 3. Bounded mapping (into a metric space): X, 3, 1. Bounded set (in a metric space): IX, 2, 2. Bounded set in $\mathbf{R}^n$: VI, 1, 1. Box (closed, open): VI, 1, 1. Broken line: VI, 1, Exercise 6.

Canonical basis of $\mathbf{R}^n$: VI, 1, 3. Capacitable set: IX, 6, 9. Capacity: IX, 6, 9. Central projection: VI, 2, 3. Centre of a ball: IX, 2, 2. Centre of a ball or sphere: VI, 2, 3 and IX 2, 2. Circle: VI, 2, 3. Class of a Borel mapping: IX, 6, Exercise 16. Closed ball: IX, 2, 2. Collectively normal space: IX, 4, Exercise 18. Compact convergence, topology of: X, 1, 3.

Compact convergence, uniformity of : X, 1, 3. Compact-open topology : X, 3, 4. Compactification, Stone-Čech : IX, 1, Exercise 7. Compatible (metric and topology) : IX, 2, 5. Compatible (metric and uniformity) : IX, 2, 4. Compatible (norm and algebra structure) : IX, 3, 7. Completely normal space : IX, 4, Exercise 3. Completely regular filter : IX, 1, Exercise 8. Completely regular space : IX, 1, 5. Completely regular space associated with a uniformizable space : IX, 1, Exercise 4. Completely separated (closed sets in a completely regular space) : IX, 1, Exercise 11. Complex hyperplane in C^n : VIII, 4, 1. Complex linear variety in C^n : VIII, 4, 1. Complex lines in C^n : VIII, 4, 1. Complex number : VIII, 1, 1. Complex number space of n dimensions : VIII, 4, 1. Complex planes in C^n : VIII, 4, 1. Complex projective line : VIII, 4, 3. Complex projective plane : VIII, 4, 3. Complex projective space of n dimensions : VIII, 4, 3. Conjugate of a complex number : VIII, 1, 1. Continuous partition of unity : IX, 4, 3. Convergent infinite product (in a normed algebra) : IX, Appendix, 3. Convergent, normally : X, 3, 2. Convergent, pointwise : X, 1, 3. Convergent, uniformly : X, 1, 1. Coordinate axis : VI, 1, 4. Coordinate variety : VI, 1, 4. Cosine of an angle : VIII, 2, 2. Cosine of a number : VIII, 2, 4. Cotangent of an angle : VIII, 2, 2. Cotangent of a number : VIII, 2, 4. Countable type (metrizable space) : IX, 2, 8. Countably compact space : IX, 2, Exercise 14. Covering, divisible : IX, 4, Exercise 16. Covering, even : IX, 4, Exercise 16. Covering, point-finite : IX, 4, 3. Cross of a pair of lines : VIII, 2, 6. Cross, right : VIII, 2, 6. Cube : IX, 1, 5. Cube (open, closed) : VI, 1, 1. Curve, Peano : VI, 1, Exercise 2.

Degree (unit of angular measure) : VIII, 2, 3. Diameter : IX, 2, 2. Diametral hyperplane : VI, 2, 4. Diametrically opposite points of $S_n$ : VI, 3, 1. Dimension of a closed subgroup of $\mathbf{R}^n$ : VII, 1, 2. Dini’s theorem : X, 4, 1. Direction ratios of a line or ray : VI, 1, 4. Direction vector of a line or ray : VI, 1, 4. Disc (open, closed) : VI, 2, 3. Discrete family of subsets : IX, 4, Exercise 18. Displacement, Euclidean : VI, 2, 2. Distance between two sets : IX, 2, 2. Distance, Euclidean : VI, 2, 1. Distance from a point to a set : IX, 2, 2. Divisible covering : IX, 4, Exercise 16. Division ring of quaternions : VIII, 1, 4. Division ring, valued : IX, 3, 2.

$\varepsilon$-period (of a function) : X, 3, Exercise 28. Equicontinuous set of mappings : X, 2, 1. Equicontinuous at a point : X, 2, 1. Equicontinuous, uniformly : X, 2, 1. Equipartition mod 1 : VII, 1, Exercise 14. Equivalent absolute values : IX, 3, 2. Equivalent families of pseudometrics : IX, 1, 2. Equivalent norms : IX, 3, 3. Equivalent pseudometrics : IX, 1, 2. Equivalent semi-absolute values : IX, 3, Exercise 10. Essential mapping into $P_n$ : VI, 3, Exercise 2. Essential mapping into $S_1$ : VI, 2, Exercise 6. Euclidean ball (closed, open) : VI, 2, 3. Euclidean displacement : VI, 2, 2. Euclidean distance : VI, 2, 1. Euclidean norm in $\mathbf{R}^n$ : VI, 2, 1. Euclidean sphere : VI, 2, 3. Even covering : IX, 4, Exercise 16. Exponential function : V, 4, 1.

Factors of a product (in a normed algebra) : IX, Appendix, 1. Family, absolutely summable : IX, 3, 6. Family of functions subordinate to a family of subsets : IX, 4, 3. Farey series : VII, 1, Exercise 13. Field of complex numbers : VIII, 1, 1. Filter, completely regular : IX, 1, Exercise 8.

Filter, maximal completely regular : IX, 1, Exercise 8. Finite open coverings, uniformity of : IX, 4, Exercise 17. Flat angle : VIII, 2, 2. Flat angular sector : VIII, 2, 5. Function, almost-periodic : IX, 3, Exercise 28. Function, exponential : V, 4, 1. Function, periodic (defined on $\mathbf{R}^n$) : VII, 1, 6. Function, $q$-ply periodic : VII, 1, 6.

Generated by a set of subsets ($\sigma$-algebra) : IX, 6, 3. Grade (unit of angular measure) : VIII, 2, 3. Group, additive (of $\mathbf{R}^n$) : VI, 1, 2. Group, additive (of real numbers mod $a$) : V, 1, 2. Group, metrizable : IX, 3, 1. Group of Euclidean displacements : VI, 2, 2. Group, one-parameter : V, 3. Group, orthogonal : VI, 2, 2.

Half-line : VI, 1, 4. Half-spaces (open, closed) determined by a hyperplane : VI, 1, 4. Hausdorff semi-absolute value : IX, 3, Exercise 9. Hausdorff semi-absolute value associated with a semi-absolute value : IX, 3, Exercise 9. Hemisphere (open, closed) : VI, 2, 4. Homographic function : VI, 3, 5. Hyper-real maximal ideal : X, 4, Exercise 16. Hyperplane at infinity : VI, 3, 3. Hyperplane, complex (in $\mathbf{C}^n$) : VIII, 4, 1. Hyperplane, diametral : VI, 2, 4. Hyperplane of projection (in a stereographic projection) : VI, 2, 4.

Imaginary axis : VIII, 1, 2. Imaginary part of a complex number : VIII, 1, 1. Imaginary, pure : VIII, 1, 1. Improper absolute value : IX, 3, 2. Inequality, triangle : VI, 2, 1, and IX, 1, 1. Inessential mapping into $\mathbf{P}_n$ : VI, 3, Exercise 2. Inessential mapping into $S_1$ : VI, 2, Exercise. 6. Infinite product : IX, Appendix, 3. Invariant metric : IX, 3, 1. Isometric mapping : IX, 2, 2. Isometry : IX, 2, 2.

Lattice in $\mathbf{R}^n$ : VII, 1, 1. Left-invariant metric : IX, 3, 1. Left-invariant pseudometric : IX, 3, Exercise 1. Length of a segment in $\mathbf{R}^n$ : VI, 2, 1. Line, broken : VI, 1, Exercise 6. Line, simple broken : VI, 1, Exercise 9. Line, complex projective : VIII, 4, 3. Line, real projective : VI, 3, 1. Linear combination of functions with coefficients in a normed space : X, 4, 4. Linear variety, complex (in $\mathbf{C}^n$) : VIII, 4, 1. Linearly accessible : IX, 6, Exercise 11. Lines, complex (in $\mathbf{C}^n$) : VIII, 4, 1. Locally paracompact space : IX, 4, Exercise 27. Logarithm to base $a$ : V, 4, 1. Logarithmic spiral : VIII, 3, Exercise 5. Lusin space : IX, 6, 4.

Mapping, almost open : IX, 5, Exercise 24. Mapping, Borel : IX, 6, Exercise 16. Mapping induced by a sifting : IX, 6, 5. Mapping into $P_n$, essential or inessential : VI, 3, Exercise 2. Mapping into $S_1$, essential or inessential : VI, 2, Exercise 6. Mapping, isometric : IX, 2, 2. Mapping, piecewise linear : VI, 1, Exercise 6. Maximal completely regular filter : X, 1, Exercise 8. Meagre set : IX, 5, 2. Measure of an angle : VIII, 2, 3. Measure of a cross : VIII, 2, 6. Measure, principal (of an angle) : VIII, 2, 3. Measure, principal (of a cross) : VIII, 2, 6. Metacompact space : IX, 4, Exercise 25. Metric : IX, 2, 1. Metric associated with a pseudometric : IX, 2, 1. Metric compatible with a topology : IX, 2, 5. Metric compatible with a uniformity : IX, 2, 4. Metric, left (right) invariant : IX, 3, 1. Metric space : IX, 2, 1. Metrizable topological group : IX, 3, 1. Metrizable topological space : IX, 2, 5. Metrizable topological space of countable type : IX, 2, 8. Metrizable uniform space : IX, 2, 4. Metrizable uniformity : IX, 2, 4. Multipliable sequence (in a normed algebra) : IX, Appendix, 1.

Nagata-Smirnov theorem : IX, 4, Exercise 22. Negative real semi-axis : VIII, 1, 2. Non-meagre space : IX, 5, Exercise 7. Norm (on a vector space) : IX, 3, 3. Norm, algebraic (of a complex number) : VIII, 1, 1. Norm compatible with an algebra structure : IX, 3, 7. Norm, Euclidean (in $\mathbf{R}^n$) : VI, 2, 1. Normal space : IX, 4, 1. Normally convergent series : X, 3, 2. Normed algebra : IX, 3, 7. Normed space : IX, 3, 3. Norms, equivalent : IX, 3, 3. Nowhere dense set : IX, 5, 1. Number, algebraic : VIII, 1. Exercise 2. Number, complex : VIII, 1, 1. Number space of dimension $n$, complex : VIII, 4, 1. Number space of dimension $n$, real : VI, 1, 1.

Obtuse angular sector : VI, 2, 5. One-parameter groups : V, 3. Open ball : IX, 2, 2. Opposite rays : VI, 1, 4. Ordered partition : IX, Appendix, Exercise 2. Ordinate : VI, 1, 4. Ordinates, axis of : VI, 1, 4. Origin of a ray : VI, 1, 4. Orthogonal affine linear varieties : VI, 2, 2. Orthogonal group : VI, 2, 2. Orthogonal transformation : VI, 2, 2. Orthogonal vectors, vector subspaces : VI, 2, 2. Oscillation of a function at a point : IX, 2, 3. Oscillation of a function in a set : IX, 2, 3.

$p$-adic absolute value : IX, 3, 7. Parallelootope (open, closed) : VI, 1, 3. Partition of unity, continuous : IX, 4, 3. Partition, ordered : IX, Appendix, Exercise 2. Peano curve : VI, 1, Exercise 2. Perfectly normal space : IX, 4, Exercise 7. Periodic function, defined on $\mathbf{R}^n$ : VII, 1, 6. Periods of a periodic function : VII, 1, 6. Piecewise linear mapping : VI, 1, Exercise 6. Pigeon-hole principle : VII, 1, Exercise 11. Plane, complex projective : VIII, 4, 3.

Plane, real projective : VI, 3, 1. Planes, complex (in $\mathbf{C}^n$) : VIII, 4, 1. Point at infinity : VI, 3, 3. Point-finite covering : IX, 4, 3. Point of uniform convergence : X, 1, Exercise 9. Pointwise convergence in a subset of $X$ : X, 1, 3. Pointwise convergence, topology of : X, 1, 3 and 3, 4. Pointwise convergence, uniformity of : X, 1, 3. Pointwise convergent (filter) : X, 1, 3. Polish space : IX, 6, 1. Polynomial in functions belonging to a given set : X, 4, 2 and 4, 4. Polynomial, trigonometric : X, 4, 4. Positive real semi-axis : VIII, 1, 2. Precompact convergence, uniformity of : X, 1, 3. Principal measure of an angle : VIII, 2, 3. Principal measure of a cross : VIII, 2, 6. Principal system of periods of a $q$-ply periodic function : VII, 1, 6. Product, absolutely convergent (of complex numbers) : VIII, 3, 3. Product, infinite : IX, Appendix, 3. Product of a multipliable sequence in a normed algebra : IX, Appendix, 1. Product, scalar : VI, 2, 2. Projection, central : VI, 2, 3. Projection, hyperplane of : VI, 2, 4. Projection, stereographic : VI, 2, 4. Projection, vertex of : VI, 2, 4. Projective space of dimension $n$, complex : VIII, 4, 3. Projective space of dimension $n$, real : VI, 3, 1. Proper at a point (group of operators) : X, 2, Exercise 18. Pseudo-compact space : IX, 1, Exercise 22. Pseudometric : IX, 1, 1. Pseudometric, invariant : IX, 3, Exercise 1. Pseudometrics, equivalent : IX, 1, 2. Pure imaginary (complex number) : VIII, 1, 1.

$q$-ply periodic function on $\mathbf{R}^n$ : VII, 1, 6. Quadric (in $\mathbf{P}_n$) : VI, 3, Exercise 10. Quadric (in $\mathbf{R}^n$) : VI, 2, Exercise 10. Quadric cone (in $\mathbf{P}_n$) : VI, 3, Exercise 11. Quadric cone (in $\mathbf{R}^n$) : VI, 2, Exercise 11. Quaternion : VIII, 1, 4.

Radian, measure : VIII, 2, 3. Radius of a ball : IX, 2, 2. Radius of a ball or sphere : VI, 2, 3 and IX, 2, 2.

Rational rank of a subgroup of $\mathbf{R}^n$ : VII, 1. Ratios, direction : VI, 1, 4. Ray (closed, open) : VI, 1, 4. Rays, opposite : VI, 1, 4. Real axis : VIII, 1, 2. Real-compact space : X, 4, Exercise 17. Real compactification of a space : X, 4, Exercise 17. Real linear variety (in $\mathbf{C}^n$) : VIII, 4, 1. Real maximal ideal : X, 4, Exercise 16. Real number space of $n$ dimensions : VI, 1, 1. Real part of a complex number : VIII, 1, 1. Real projective line : VI, 3, 1. Real projective plane : VI, 3, 1. Real projective space of $n$ dimensions : VI, 3, 1. Real valuation of a division ring : IX, 3, 2. Rectangle (open, closed) : VI, 1, 1. Re-entrant angular sector : VIII, 2, 5. Right angle, positive : VIII, 2, 2. Right angular sector : VIII, 2, 5. Right cross : VIII, 2, 6. Right-invariant metric : IX, 3, 1. Right-invariant pseudometric : IX, 3, Exercise 1.

$\sigma$-algebra : IX, 6, 3. Salient angular sector : VIII, 2, 5. Saturated family of pseudometrics : IX, 1, 2. Saturated set of subsets : X, 1, Exercise 5. Scalar product : VI, 2, 2. Sector, angular : VIII, 2, 5. Segment (closed, open, open at $x$ and closed at $y$) : VI, 1, 4. Segment, length of : VI, 2, 1. Semi-absolute value : IX, 3, Exercise 9. Semi-absolute value, Hausdorff : IX, 3, Exercise 9. Semi-absolute values, equivalent : IX, 3, Exercise 10. Semi-axis, negative real : VIII, 1, 2. Semi-axis, positive real : VIII, 1, 2. Semicircle (open, closed) : VI, 2, 4. Separable metrizable space : IX, 2, 8. Separating set of functions : X, 4, 1. Separation of points of a set by a set of functions : X, 4, 1. Sequence, multipliable : IX, Appendix, 1. Series, absolutely convergent (of points of $\mathbf{R}^n$) : VII, 3, 2. Series, Farey : VII, 1, Exercise 13. Set, almost open : IX, 5, Exercise 6.

Set, Borel : IX, 6, 3. Set, bounded : IX, 2, 2. Set, bounded (in $\mathbf{R}^n$) : VI, 1, 1. Set, capacitable : IX, 6, 9. Set, meagre : IX, 5, 2. Set, nowhere dense : IX, 5, 1. Set, Souslin : IX, 6, 2. Set, starlike (in $\mathbf{R}^n$) : VI, 2, Exercise 12. Set, thin : IX, 5, Exercise 2. Shell of a starlike set : VI, 2, Exercise 12. Side of a cube : VI, 1, 1. Sides of a broken line : VI, 1, Exercise 6. Sieve : IX, 6, 5. Sifting : IX, 6, 5. Sifting, mapping induced by : IX, 6, 5. Simple broken line : VI, 1, Exercise 9. Sine of an angle : VIII, 2, 2. Sine of a number : VIII, 2, 4. Slice (in a linearly ordered set) : IX, Appendix, Exercise 2. Slope of a line : VIII, 2, 6. Souslin set : IX, 6, 2. Souslin space : IX, 6, 2. Space associated with a sifting : IX, 6, 5. Space, Baire : IX, 5, 3. Space, collectively normal : IX, 4, Exercise 18. Space, completely normal : IX, 4, Exercise 3. Space, completely regular : IX, 1, 5. Space, complex number : VIII, 4, 1. Space, complex projective : VIII, 4, 3. Space, countably compact : IX, 2, Exercise 14. Space, locally paracompact : IX, 4, Exercise 27. Space, Lusin : IX, 6, 4. Space, metacompact : IX, 4, Exercise 25. Space, metric : IX, 2, 1. Space, metrizable of countable type : IX, 2, 8. Space, metrizable topological : IX, 2, 5. Space, metrizable uniform : IX, 2, 4. Space, non-meagre : IX, 5, Exercise 7. Space, normal : IX, 4, 1. Space, normed : IX, 3, 3. Space of loops : X, 3, 4. Space of paths : X, 3, 4. Space of projective linear varieties of dimension $p$ in complex projective space of dimension $n$ : VIII, 4, 3.

Space of projective linear varieties of dimension $p$ in real projective space of dimension $n$: VI, 3, 5. Space, perfectly normal : IX, 4, Exercise 7. Space, Polish : IX, 6, 1. Space, pseudo-compact : IX, 1, Exercise 21. Space, real-compact : X, 4, Exercise 17. Space, real number : VI, 1, 1. Space, real projective : VI, 3, 1. Space, separable metrizable : IX, 2, 8. Space, Souslin : IX, 6, 2. Space, strongly zero-dimensional : IX, 6, Exercise 1. Space, submetrizable : IX, 2, Exercise 23. Space, totally non-meagre : IX, 5, Exercise 12. Space, ultrametric : IX, 2, Exercise 4. Space, zero-dimensional : IX, 6, 4. Sphere : IX, 2, 2, VI, 2, 3. Spiral, logarithmic : VIII, 3, Exercise 5. Square (closed, open) : VI, 1, 1. Starlike set in $\mathbf{R}^n$ : VI, 2, Exercise 12. Stereographic projection : VI, 2, 4. Stone-Čech compactification : IX, 1, Exercise 7. Stone’s theorem : X, 4, 1. Stone-Weierstrass theorem : X, 4, 2. Strict sifting : IX, 6, 5. Strongly zero-dimensional space : IX, 6, Exercise 1. Subgroup, associated : VII, 1, 3. Submetrizable space : IX, 2, Exercise 23. Subordinate to a family of subsets (family of functions) : IX, 4, 3. Support of a function : IX, 4, 3.

Tangent of an angle : VIII, 2, 2. Tangent of a cross : VIII, 2, 6. Tangent of a number : VIII, 2, 4. Theorem, Ascoli’s : X, 2, 5. Theorem, Baire’s : IX, 5, 3. Theorem, Dini’s : X, 4, 1. Theorem of R. Ellis : X, 3, Exercise 25. Theorem of Nagata-Smirnov : IX, 4, Exercise 22. Theorem, Stone’s : X, 4, 1. Theorem, Stone-Weierstrass : X, 4, 2. Theorems, Urysohn’s : IX, 4, 1 and 4, 2. Thin set : IX, 5, Exercise 2. Topology, compact-open : X, 3, 4. Topology of compact convergence : X, 1, 3 and 3, 4.

Topology of pointwise convergence : X, 1, 3. Topology of pointwise convergence in a subset of X : X, 1, 3. Topology of S-convergence : X, 1, 2. Topology of uniform convergence : X, 1, 1. Topology of uniform convergence in a subset of X : X, 1, 3. Torus, one-dimensional : V, 1, 2. Torus, n-dimensional : VII, 1, 4. Torus of revolution : VII, 1, Exercise 15. Totally non-meagre space : IX, 5, Exercise 12. Transformation, orthogonal : VI, 2, 2. Triangle inequality : IX, 1, 1. Triangle inequality : VI, 2, 1. Trigonometric form of a complex number : VIII, 2, 2. Trigonometric polynomials in m variables : X, 4, 4.

Ultrametric space : IX, 2, Exercise 4. Uniform approximation of a continuous real-valued function by continuous functions belonging to a given set : X, 4, 1. Uniform convergence in a subset of X : X, 1, 3. Uniform convergence in the sets of S : X, 1, 2. Uniform convergence, point of : X, 1, Exercise 9. Uniform convergence, topology of : X, 1, 1. Uniform convergence, uniformity of : X, 1, 1. Uniformity, additive (of R^n) : VI, 1, 2. Uniformity of bounded convergence : X, 1, 3. Uniformity of compact convergence : X, 1, 3. Uniformity defined by a family of pseudometrics : IX, 1, 2. Uniformity defined by a pseudometric : IX, 1, 2. Uniformity of finite open coverings : IX, 4, Exercise 17. Uniformity of pointwise convergence : X, 1, 3. Uniformity of pointwise convergence in a subset of X : X, 1, 3. Uniformity of precompact convergence : X, 1, 3. Uniformity of S-convergence : X, 1, 2. Uniformity of uniform convergence : X, 1, 1. Uniformity of uniform convergence in a subset of X : X, 1, 3. Uniformity of uniform convergence in the sets of S : X, 1, 2. Uniformity, universal : IX, 1, Exercise 5. Uniformly convergent (filter) : X, 1, 1. Uniformly convergent in every set of S (family of mappings, filter, sequence, series) : X, 1, 2. Uniformly equicontinuous set of mappings : X, 2, 1. Uniformly summable family : X, 1, 2. Unit ball, sphere : VI, 2, 3 and IX, 3, 3. Unit of angular measure : VIII, 2, 3.

Universal uniformity : IX, 1, Exercise 5. Urysohn’s theorems : IX, 4, 1 and 4, 2.

Valuation, real : IX, 3, 2. Value, absolute : IX, 3, 2. Value, absolute (of a complex number) : VIII, 1, 1. Value, semi-absolute : IX, 3, Exercise 9. Valued division ring : IX, 3, 2. Variety, coordinate : VI, 1, 4. Vector, direction (of a line or ray) : VI, 1, 4. Vector subspaces, orthogonal : VI, 2, 2. Vectors, orthogonal : VI, 2, 2. Vertex of a stereographic projection : VI, 2, 4.

Weierstrass-Stone, theorem of : X, 4, 2.

Zero-dimensional space : IX, 6, 4