 83
 71 72 75 83 143
Accumulation point 133—139 151
Addition 28 33—36 55 60 94
Adherence point 12 13 17 20—23 29 60 65—68 74 75 78 105 106 109 110 133 149
Adherence set 66 67
Aftine transformation 28 29 44 65 94 97
Antidiagonal 45 139
Axioms of topology 9—11
Basic neighbourhood 16 20 28
Bijection 1 27 29 30 52 76
Bilinear function 35
Binary expansion 142 143
Binary operation 25 36
Bounded function 70 154
Bounded metric 59 90
Bounded set 59 61 64 65 115 119 157
Bounded uniformity 100
Bourbaki, N. 62 76
Box topology 31 39 70
Cartesian product 3 30 38 67 92
Cauchy filter 107—112 148—157
Cauchy sequence 107—109 112 150—155
Centre 128
circle 47 56 119
Closed ball 19 20 21 64 76 118 122
Closed covering 45 46 60 112
Closed function 53 56—64 68 70 73—76 131
Closed neighbourhood 16 72 73 76 78 81—84
Closure 12 13 20—23 27 32 38—40 51 52 57 66—68 74 77—79 81—83 89 99 103—105 116—118 132 133 141 144 145 154—157
Closure point 12
Cocountable topology 29
Cofinite topology 5 8 11 12 23 26 30 40 63 80 105 114 128
Coherent system 17—20 66 101
Coincidence set 74 116
Coinduced topology 48—50
Coinduced uniformity 98
Collapsing 50 76 79
Collins, P.J. 126
Commutative 5 74 88 91 157
Compact space 62—70 76—85 105—107 113 115 127—131 136 139 146 147 153
Compactly regular 82 83 157
Complete family 14 30 32 55 78 130
Complete metric space 150 151 154—157
Complete uniform space 148—157
Completely regular 140—143 146
complex numbers 30 35 47
composition 4 26 54 87 91
Conjugation 37 38 120
Connected space 114—117 120 121 128
Connectedness component 118—120 123 124
Constant function 26 40 70 90 114—117 125 128
Continuous function 26—61 63 65 70 73—75 79 80 102 107 114—125 130—137 140—149
Convergent filter 22 28 32 67 78 108 148 149 152
Convergent sequence 21 22 29 107—109 134—138 153
Convex subspace 122
Countable set 8 23 52 69 128 136—138
Countably compact 132—139
Covering 45 57 60 64 68 69 open
Cylinder 76
Dense 12 23 40 51 52 74 105 116 117 136—138 149 152—156
Determinant 35 48 56
Diagonal 3 31 36 39 43 45 47 51 73 85—90 93 96 98 101—103 106 113 126 143
Diameter 59 61 90
Dilatation 28 31 35 94
Direct product 38 39 51 94
Disconnected 115 122
Discrete group 50 88 91 128
Discrete metric 18 20 59 89 138
Discrete topology 9—14 20—26 29 30 36 39 44 45 48—53 57 62 63 77 79 82—84 101 102 113—117 120 121 139
Discrete uniformity 86—89 90 100 101 109 125 126 148
Division 38 47 52
Dual family 11 68
Dyadic expansion 142—144
Elementary filter 5—8 22 109—112 136 150
Embedding 41—44 52 54 57 63 64 73—77 80 119 131 141 143
Entourage 85—113 125—127
Equivalence class 4 36 50 117
Equivalence relation 4 39 50—52 76 79 83 89 98 100 117 118 123 124 128 147 155
Euclidean metric 18—21 59 84 90 100
Euclidean topology 20 36 39 44 46 52 58—61 64 77 84 115 118 125 128 130 157
Euclidean uniformity 87 88 91 94 97 100 107 108 112 126
Exponential function 97 100 108
Factor group 55 56 98 125
Factor space 51 55 75—78
Filter 5—8 21—23 28 39 65—68 85—87 109
Filter base 6 8 16 19 87 88 106 113 128
Final topology 41
Finite set 8 11 26 63 89 98 100 108 110 139
First countable 130 135—139
Function space 67 111
Functionally Hausdorff 140 147
Functionally normal 140
Functionally regular 140
Functionally separated 140—147
Fundamental system of neighbourhoods 18 101
General linear group 48 65
Generating family 13—14 19 27—32 41—46 55 70 86 120 131 137 138 141
Graph 43 64 73 75 83 96
Hausdorff quotient space 79 105
Hausdorff space 62 72—85 103 106 113 127—129 133 136—139 147 157
Heine — Borel theorem 64 65 153
Hilbert sequence space 19 150
Homeomorphism 29—31 34 35 45—52 56 62 76 79 102 119 125 141
Homomorphism 25 37—39 44 91 98
Idempotent 12
Identification space 50
Identity relation 4
Indexing set 1 3 33 39 93
Indiscrete topology 10
Induced topology 41—44 48 50 95 103
Induced uniformity 95 102 110
Infimum 20 65 115 143 145
Infinite set 11 21 105 113 133 136 151
Initial topology 41 42
Injection 1 24 42 43 54 57 73 95 96 156
Interior 12 13 23 32 39 40 54 83 104
Interior function 53
Interior point 12 13 17 119
Intermediate Value Theorem 116
interval 14—16 27 44—46 50 56 59 83 97 116 119
Interval topology 15 16 27 44
inversion 36 46—48 51 55 91 94 157
Irrationals 16
Isometry 28 153—155
Isomorphism 37 56
Juxtaposition 123
Kelley, J.L. 83
Lebesgue number 69
Left inverse 43 96
LIMIT 21 22 153
Limit point 22 23 29 32 75 109 110 134 150—153
Lindelof 131 132 139
Linearly ordered set 14
Locally connected 120 121 124—128
Locally pathwise-connected 124 125
Lower central series 39
Lower limit topology 15
Lower Sorgenfrey topology 15
Matrices 30 34 35 48 94
Matrix multiplication 35 48
Maximum 46 94
Metric 18 24 28 33 59 88 90 155
Metric completion 154 155
Metric space 18—21 27 28 45 65 69—71 77 80 85 91 101 102 108 126 137 138 150—155
| Metric topology 20 33 59 71
Metric uniformity 88
Minimal Cauchy filter 112 113
Minimum 33 46 59 69 90 94 151
Modular group 56
Multiple induced topology 42
Multiplication 28 34 35 47 55 61 91 94 137 138
Negative 36 94
Neighbourhood 13 16 17 21 23 51 66 70—73 78 79 83 100
Neighbourhood base 16 17 21 22 32 33 38 39 45 72 82 88 101—105 120 124 126 128 135—138
Neighbourhood filter 16 17 20—23 28 29 33 34 45 59 65 75 76 88 102 109 113 136 152
Neighbourhood of diagonal 36 106 107 113
Nesting condition 8 151
Net 90 108 151
Norm 33
Normal space 80—82 84 132 144—147
Normal subgroup 38—40 55—56 98 119 120 125 128 157
Open ball 19—21 47 59 88 101 108 118 122 138
Open covering 45 46 60 68 69 81 113 128 131—134 151
Open function 53—56 75 130
Open neighbourhood 13 16 26—28 34 37—40 51 56—60 63 70—73 77—83 120 124 128 132 140 141
Orbit space 50
Order 14
Order-preserving function 27
Ordered set 14 15 27 44
Orthogonal group 65 76
p-adic structure 88 128
Partially ordered set 14
Partition 102 118
Path 121—125
Path-component 123—125
Pathwise-connected 121—125
Pointwise complete 157
Pointwise convergence 112
Pointwise topology 67
Polar coordinates 76
Polynomial function 35 39
Precompact 89
Principal filter 5 7 16 20 21 66 109 155
Product neighbourhood 32
Product set 30 31
Product topology 30—33 38 39 45 63 67 103 104 131
Product uniformity 93 98 103 111
Projective space 50 52
Proper function 70
Property S 126 127
Punctured circle 119
Punctured complex line 47 158
Punctured open ball 21
Punctured real line 46 94 97 118
Punctured sphere 64
Quasicompact 62
Quotient map 49—57 82 121 128
Quotient space 50—52 75 76 79 124 128 147
Quotient topology 50 51 55 117
Quotient uniformity 98
Rational line 16 29 46 47 77 82 112 115 118 125 128 136 157 158
Ray 15
Real line 4 16 18 20 23 24 31 36 39 46 47 50 53 56 59 61 64 65 68 77 82 84 87 88 116—120 124
Real rc-space 18 21 36 46 47 59 64 65 88 118—121 124 146
refinement 6 7 9 13 21—23 67 77 109 112 152
Reflexive 4 36 78 79 87 89 118
Regular space 76—83 89 104—106 132 147
Regularly closed 23
Regularly open 23
Relation 3—5 36 83 88 100
Relative topology 44—46 51—54 65 67 103 139
Relative uniformity 97 98 103 105
Restricted product neighbourhood 32 78 141
Restricted product set 30 32 93 117 120
Reverse path 123
Reverse relation 4
Right inverse 50 52
Saturation 2 55 57 78 79
Scalar division 47
Scalar multiplication 35 47
Second countable 120 130—132 136 138
Separable 136—139
Separated quotient space 98 99 105 152 156
Separated uniform space 89 94 103 106 112 148 152—157
Separation conditions 71
SEQUENCE 5 19—22 33
Sequentially compact 134—136 139
Sequentially complete 150 151
Sierpinski 10 20 30 39 80 121 126
Small 89 109—112 126 127 151 156
Sorgenfrey line 15 44 139
Sorgenfrey plane 44 139
Sphere 21 47 50 64 76 100 118 122
Squaring function 35 53 57 91
Stationary path 123
Stereographic projection 47
Structure-preserving function 25 53 90
Subcovering 45 68 131 132
Subgroup 38 39 45 51 52 55 70 74—78 98 117 128
Subsequence 5 67 108—110 134 135
subspace 41 44—48 64 73 80 97 98
Supremum metric 19 65 154
Surjection 1 49 53 54 63 76 80 99 115 121 124 125 130 133 137
Surrounding 85
symmetric 4 36 78 87—89 99 103—105 118 127
Symmetric neighbourhood 38 39
Tietze theorem 145 146
Topological equivalence 29
Topological group 36—40 45 55 64 75 78 82 85 88—91 94 98—102 105 107 117 119 125 126
Topological product 30—36 39 42 55 65 73 80 103 117 120 121 131—134 137 140 141
Topological space 9—84 101—158
Topologists'sine curve 122
Topology of pointwise convergence 67
Topology of uniform convergence 111
Totally bounded uniformity 89 90 94 98 108—111 127 137 138 151 153 157
Totally disconnected 115 118 128 129
Trace 6 44—46 66 82 97 105 106 152
Transitive 4 36 42 49 78 79 86 89 95 123
Translation 28—31 34 37 51 94 120
Trivial topology 10—12 22—26 30 36 42—44 48 51—53 57 71 77—80 101 114 120 128
Trivial uniform space 86—90 96 101 109 148
Tychonoff theorem 64 67 68
Ultrafilter 7 8 23 67 110—113
Uncountable 40
Uniform completion 155—158
Uniform embedding 95—97 155 156
Uniform equivalence 92 97 102 153—156
Uniform neighbourhood 101—109 113 126
Uniform product 92—97 100 125 149 157
Uniform space 85—113 142 148 155
Uniform topology 101—104 113 142
Uniformity base 87—90 96 101 104—107 126 156
Uniformity of pointwise convergence 111
Uniformity of uniform convergence 111 149
Uniformization 106 107 111 141—143
Uniformly Cauchy 112 149
Uniformly complete 149 150
Uniformly connected 125 126
Uniformly continuous function 90—100 102 107 110 112 125 143
Uniformly convergent 112
Uniformly homogeneous 92 93
Uniformly locally connected 127
Uniqueness 22
Unit circle 18
Upper limit topology 15
Upper Sorgenfrey topology 15 44
Urysohn theorem 144 145
Weak transitivity 88 98 99
Whyburn, G.T. 126
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