(= of uniform convergence on members of 
), 228
Absolute Gδ, 207
accumulation point, 41
from the right, 59
ω ..., 137
adjoint of a normed linear space, 241
Alexander subbase theorem, 139
almost periodic function, 247
amalgamation axiom, 261
anti-symmetric relation, 9
asymmetric relation, 263
Baire, category theorem, 200
condition of … (= almost open), 210
Banach algebra, 243
base, for the neighborhood system of a point, 50
for the neighborhood system of a set, 112
for a topology, 46
for a uniformity, 177
bicompact (= compact), 135
σ-ring, 215
boundary of a set, 45
bounded set, 144
Brouwer reduction theorem, 61
Cantor discontinuum (= middle third set), 165
cartesian product, of two sets, 7, 261
of a family of sets, 30
category, 201
(first category = meager)
(second category = non-meager)
Cauchy, filter, 193
net, 190
characteristic function, 26
function, 31
classification axiom scheme, 253
classifier, 251
closed, family, 155
graph theorem, 213
interval, 40
map, 94
set, 40
sphere, 119
closure, and complement problem, 57
axioms, 43
of a set, 42
operator, 43
cluster point, of a net, 71
of a set, 41
coarser (= smaller) topology, 38
compact, 135
countably, 162
locally, 146
… open topology, 221
compactification, 149 ff
one-point, 150
Stone-Čech, 152
Wallman, 167
… of groups, 247
complete, accumulation point, 163
(= full) class, 267
uniform space, 192
completely regular, 117
completion, 196
of topological groups, 211
component, 54
composition of relations, 7, 260
connected, 53
locally, 61
connects, 263
continuous, at a point, 87, 100
convergence, 241
evenly on a set, 237
function, 85
in one of two variables, 103
jointly, 223
continuum hypothesis, 280
contravari ant correspondence, 246
convex, 103
coordinate set, 31
convergence, classes, 73
continuous, 241
Moore-Smith, 62 ff
pointwise( = coordinate wise = simple), 92
sequential, 62
correspondence (= function), 10
countability, first axiom, 50
second axiom, 48
countable, 25
chain condition, 60
countably compact, 162
counter-image (= inverse image), 11
covariant correspondence, 246
cover, 49
even, 155
open, 49
point finite, 171
uniform, 199
cube, 114
decimal expansion, 25
decomposition, 96
upper semi-continuous, 98
dense, 49
nowhere, 145
denumerable (= countably infinite), 25
derived set, 42
diagonal, 7
process, 238
diameter, 121
dictionary (= lexicographic) order, 23
Dini’s theorem, 239
directed set, 65
directs, 65
discrete, topological space, 37
family, 127
σ-discrete family, 127
disjoint, 3
distance, 119
between sets, 123
Hausdorff, 131
distinguishes, functions, 220
points, 116
points and closed sets, 116
divisible properties, 133
door space, 76
dyadic, expansion, 25
rational, 23
embedding, in cubes, 114 ff
equicontinuous, 232
on a set, 234
uniformly, 239
equipollent, 28
equivalence, class, 9
relation, 9
Euclidean, plane, 60
continuity of …, 240
continuity on a set, 237
cover, 155
eventually, 65
extension, of a function, 11
Tietze theorem, 242
extent axiom, 252
extrem ally disconnected, 216
field, 18
ordered, 19
filter, 83
finer (= larger) topology, 38
finite, 277
character, 32
intersection property, 135
frequently, 65
full, 267
gage of a uniformity, 189
group, 17
abelian (= commutative), 17
normal (= distinguished = invariant), 18
quotient (= factor), 18
half-open, interval, 40
Hausdorff, maximal principle, 31, 274
metric, 131
topological space, 67
uniform space, 180
hereditary properties, 133
homeomorphism, 87
homogeneous space, 107
hypothesis of the continuum, 280
ideal, 18
dual, 80
lattice, 80
identity relation, 7
indiscrete (= trivial) topological space, 37
induction, definition by, 21
transfinite, 270
inductive, property (= satisfying premise of 0.25 a), 33
set of integers, 20
infimum, 13
infinite cardinals, 280
infinity axiom, 271
integration, 77 ff
interior, of a set, 44
map, 90
inverse, of a relation, 7
image, 11
irreducible property, 61
isolated point x (= {x} is open), 102
isometry, 122
of compact metric space, 162
isotone function, 15
iterated, limit, 69
sum, 77
join, of sets ( = union), 3
of members of a lattice, 80
joint continuity, 221
k-extension of a topology, 241
Kuratowski, closure axioms, 43
closure and complement problem, 57
lemma, 33
largest, 31
lattice, 80
lexicographic (= dictionary) order, 23
limit, iterated, 69
of a net, 68
point of a set, 41
Lindelöf, theorem, 49
linear, function, 18
order (= simple order), 14
topological space, 109
locally, compact, 146
connected, 61
finite, 126
lower bound, 13
map (= function), 10
maximal, member of a family, 32
meager (= first category), 201
meet, of sets, 3
of members of a lattice, 80
metacompact, 171
metric, 118
invariant, 123
topology, 119
uniformity, 184
Hausdorff …, 131
metrization, of topological spaces, 124 ff
of uniform spaces, 184 ff
minimal member of a family, 32
principle, 33
monotone function, 15
Moore-Smith convergence, 62 ff
neighborhood, 38
system of a point, 39
system of a set, 112
topologies from … systems, 56
nest, 32
net, 65
universal, 81
normal, topological space, 112
perfectly, 134
fully, 170
nowhere dense, 145
normed linear space, 241
null space of a linear function, 19
real, 19
one-to-one function, 11
open, interval, 38
map, 90
set, 37
sphere, 119
almost …, 210
operator (= function), 10
order, Archimedean, 22
bounded, 15
complete, 14
dictionary (= lexicographic), 23
linear (= simple = total), 14
product, 68
ordered pair, 259
ordering, partial, 13
quasi (= partial), 13
partition, of a set, 96
of unity, 171
perfectly normal, 134
point finite cover, 171
pointwise convergence, 92
on a set, 220
on a dense set, 238
uniformity of …, 220
power of a set, 275
precompact (= totally bounded), 198
directed set, 68
logical, 3
of compact spaces, 143
of connected spaces, 104
of pseudo-metric spaces, 122
of spaces with countable bases, 103
order, 68
topology, 90
uniformity, 182
productive properties, 133
projection, onto a coordinate set, 31
onto a quotient space, 96
pseudo-metric, 119
topology, 119
uniformity, 184
pseudo-metrizable, topological space, 124
uniform space, 184
quotient, group, 18
map, 96
topology, 94
real numbers, 19
topological properties, 58, 59
refinement, 128
regular topological space, 113
regularity axiom, 266
reflexive relation, 9
relative, topology, 51
uniformity, 182
relativization, 50
residual (= co-meager), 201
restriction of a relation, 10, 11
ring, 18
scalar field, 18
Schroeder-Bernstein theorem, 28, 276
section, 263
semi-metric (= pseudo-metric), 119
perfectly, 48
separated, sets, 52
topological space (= Hausdorff), 67
uniform space ( = Hausdorff), 180
separation properties, with compactness, 140 ff
sequentially compact, 162, 238
theory, 250 ff
simple topology (= topology of point-wise convergence), 90, 92, 217
small sets, 192
smallest member of a family, 32
space, completely regular, 117
Hausdorff, 67
Lindelöf, 50
linear, 18
k-space, 230
linear topological, 109
metric, 119
normal, 112
regular, 113
T0-space, 56
T1-space, 56
T2-space (= Hausdorff), 67
T3-space (= regular + T1), 113
T4-space (= normal + T1), 112
topological, 37
Tychonoff (= completely regular + T1), 117
uniform, 176
Stone-Weierstrass theorem, 244
stronger topology, 38
subadditive function, 131
subbase, for a topology, 48
for a uniformity, 177
for the neighborhood system of a point, 50
local, 50
subclass, 256
subcover, 49
subsequence, 63
subset, 2
proper, 2
axiom, 256
subnet, 70
subspace, 51
substitution axiom, 261
sum, logical, 3
supremum, 13
symmetric relation, 9
σ-compact, 172
σ-discrete, 127
σ-locally finite, 127
σ-ring, 215
T−-space : see space
Tietze extension theorem, 242
topological groups, 105
closed graph theorem, 213
completion, 211
uniformities and metrization, 210
topological, invariant, 88
map, 87
space (see also space), 37
topologically equivalent, 87
topology, 37
coarser (= smaller), 38
discrete, 37
indiscrete (= trivial), 37
finer (= larger), 38
from interior operators, 56
from neighborhood systems, 56
metric or pseudo-metric, 119
of compact convergence (=* uniform convergence on compacta), 229
order, 57
pointwise (= product), 90
quotient, 94
relative, 50
stronger, 38
uniform, 178
u.c. = of uniform convergence, 226
usual, for the real numbers, 37
weaker, 38
totally bounded, 198
tower (= nest), 32
transitive relation, 9
transfinite induction, 270
transformation (= function), 10
triadic expansion, 25
triangle inequality, 119
trivial (= indiscrete) topology, 37
two point property, 244
two set property, 243
Tukey’s lemma, 33
Tychonoff, plank, 132
product theorem, 143
space, 117
ultrafilter, 83
uniform, boundedness, 215
continuity, 180
continuity on a set, 194
convergence, 226
convergence on compacta, 229
cover, 199
covering system, 207
equicontinuity, 229
equivalence, 181
invariant, 181
isomorphism, 181
neighborhood system, 205
space, 174 ff
topology, 178
uniformity, generated by pseudo-metrics, 184, 187
of pointwise convergence, 220
of uniform convergence, 226
product, 182
relative, 182
u.c. (= of uniform convergence), 226
usual, for real numbers, 177
(= of uniform convergence on members of ), 228
uniformly, locally compact, 214
open map, 202
axiom of …, 258
universal net, 81
universe, 255
unordered pair, 258
upper bound, 13
upper semi-continuous, decomposition, 99
function, 101
Urysohn lemma, 115
metrization theorem, 125
usual, metric for real numbers, 119
topology for real numbers, 37
uniformity for real numbers, 177
value of a function, 261
vector (= linear) space, 18
void, 255
w* (= weak*) topology, 108
principle, 32
Zermelo postulate, 33
zero, 255
Zorn’s lemma, 33