Index

A B Absolutely countably , 286 Baire property, 9, 69 Accumulation point, 81 Baire space, 9–11 of a sequence of sets, 80 Banach space, 80 Action, 218, 220, 222–228, 231, 234, 241, Base for the closed sets, 133 242 Base Tree Theorem, 274 Bernstein set, 158 algebraic, 220, 223, 234, 236 b f -group, 221, 224, 226–228 b f -action, 226 b f -space, 108, 122, 124, 125, 221, 222, 224, diagonal, 222 226–228, 231, 237 effective, 232 Birkhoff–Kakutani theorem, 39 faithful, 232 b-lattice, 164–167 free, 231 Bounded linearization of, 241 lattice, 164 trivial, 218, 229, 230, 240 neighborhood, 123 AD, see almost disjoint family open subset, 110 subset, 107–113, 115–119, 121–126, Additive group of the reals, 242 128–133, 136, 137, 139–142, 144–146, Admissible , 234 158, 221, 222 Alexandroff one-point compactification, 233 Boundedness, 111 Algebra of continuous functions, 221 Bounding number, 268 Almost compact space, 16, 17, 266 br -space, 108, 122, 124, 128, 129, 137 Almost disjoint family, 7, 8, 17, 28, 36, 195, 253–261, 266, 269, 271, 272, 280– 282, 284, 286 C Almost dominance with respect to a weak Canonical η-layered family, 263 selection, 280 Canonical in a GO-space, 170, 173 Almost metrizable , 60 C-compact subset, 34, 36, 108, 115–117, Almost periodic point, 242 126–129, 137, 140, 184 ˇ Almost periodic subset, 242 Cech-complete space, 174–176, 181 Cech-completeˇ topological group, 181 Almost pseudo-ω-bounded space, 19, 20 Cellular family, 19 Almost P-space, 176 Cellularity, 49 (α, ) D -, 77, 78, 85 C-embedded subset, 2–4, 15, 16, 18, 26, 110, Arcwise connected space, 184 122, 225 Arzelà-Ascoli theorem, 108, 239, 240 C∗-embedded subset, 2, 13, 17, 24, 25 Ascoli theorem, 80 Centered family, 275 © Springer International Publishing AG, part of Springer Nature 2018 291 M. Hrušák et al. (eds.), Pseudocompact Topological Spaces, Developments in Mathematics 55, https://doi.org/10.1007/978-3-319-91680-4 292 Index

Centered sequence, 175, 178 Countable chain condition, 220 CH, see Continuum Hypothesis Countably closed family, 178 Character (topological), 229 Countably compact group, 62, 72 Circle group, 40 , 6–8, 15, 18, 35, Closed invariant subgroup, 221 97, 111–113, 115, 158, 176, 192, Closed map, 236 201–203, 205, 206, 208–214, 254, Cofinal subset, 178, 179, 181 273 Collectionwise , 112 , 115, 122 Cω-discrete space, 183 countable, 115 Y C<ω-discrete space, 183–186 subcover, of, 115 Y Cω -embedded space, 183 the cover b, 122 Comfort–Ross theorem, 40, 61 the cover c, 129 Compact group, 98, 104 the cover k, 122 Compactification, 14, 16, 30, 80, 130, 153 the cover sb, 122 Alexandroff compactification, 116 Cozero-set, 2–4, 115, 152 equivalent compactifications, 130 cr -space, 129 Stone-Cechˇ compactification, 13, 17, 33, CY -embedded subspace, 35, 183 78, 107, 109, 110, 115 Compact Lie group, 219 D Compact metrizable group, 184, 196 Dense subset of [ω]ω, 274 Compact metrizable space, 32, 34–36, 90, De Vries’ theorem, 231 112, 194, 200 Diagonal map, 178 Compact-open topology, 107, 223, 231–233 Diagonal, of a space, 56, 111, 112, 144 Compact space, 2, 6, 8, 11, 13–19, 26, 27, ♦-principles, 286 32, 34, 35, 79, 80, 89, 97, 107, 110– Dieudonné completion, 108–111, 130, 137, 113, 115, 119, 121, 122, 127, 133, 138, 218, 222, 224–229, 231 143, 192, 198, 200, 202, 204, 205, Dini theorem, 80 208, 231 Discrete collection of subsets, 6 Completely metrizable group, 182 Dispersion index 2, 199 Completely metrizable space, 155, 175 Distribution of the functor β X, 237 Completely separated sets, 115, 116 G Distribution of the functor of the topological Complete metric, 228 completion, 122 Complete sequence, 175, 179 Distributivity number, 274 Concentrated family, 269 Dominance with respect to a weak selection, Condensation, 111, 112 280 Connected group, 104 D-pseudocompact space, 85, 86, 99, 101 Connected space, 94, 214, 278 Dyadic rational number, 246 Continuous extension of a map, 220, 221, Dyadic space, 32–34, 36, 200, 213 224, 225, 227 Continuous map, 217, 218, 220, 223–225, 227, 234 E b f -continuous, 221, 224 Eberlein compact space, 204 Continuous real-valued function, 221–223, Efimov space, 39, 51 225, 237, 241, 245, 246 Equicontinuous family, 130, 133, 136, 138, bounded, 220, 221, 241 223 Continuous selection, 278 at a point, 223 Continuous weak selection, 278–280, 284 invariant subset, 223 Continuum Hypothesis, 198, 209, 258, 263, pointwise bounded, 137–139 264, 267, 271 Equivalent compactifications, 153 Convergent shrinking subsequence, 279 Equivalent layered families, 263 Corson compact space, 204 Equivariant map, see G-map Coset space, 141, 144, 221, 222, 240 Essential extension, 260 Countable cellularity, 33 Essential family, 248 Index 293

locally finite, 248, 250 C(X), 130, 133, 139 η-layered family, 263, 264 C∗(X), 124 Evaluation map, 234 C p(X), 108, 113, 115 Exponential map, 108 Functionally bounded set, 108 Extent, 272 Functional tightness, 210–212, 215

F G Family γ f -space, 122 almost disjoint, 7, 8, 17, 28, 36, 195 γr -space, 122 cellular, 19 G-compactification, 218–221, 223, 224, countable closed, 178 228–231, 233, 235, 237, 240, 241 locally finite, 3–6, 8–12, 18, 65 equivalent G-compactifications, 218, maximal almost disjoint, 7, 195 219 point-finite, 10, 11 equivariant compactification, 241 Feebly compact space, 36, 191, 194, 196, maximal, 219, 223, 236 274–276, 278 G-compactification problem, 219 Filter, 117 Gδ-closure, 115, 116 Finest uniformity, 146 Gδ-dense subset, 14, 34, 36, 44, 115, 143, Finite intersection property, 3, 4, 10, 14, 15 152, 153, 160, 163, 164, 169, 175, F.I.P., see finite intersection property 176, 182–184, 225 First countable Gδ-diagonal, 112, 174, 254 group, 39, 47, 64, 92 property, 112 space, 92, 111, 122, 192, 195, 196, 202, Gδ-set, 14, 52, 61, 115, 121, 222, 225 205, 208, 254 G-disjoint subsets, 244, 247 topology, 196 General linear group, 41 F-pseudocompact space, 99, 101 Generalized linearly ordered topological Franklin compactum, 262, 272 space, 112, 126, 140, 170–173 Fréchet-Urysohn space, 197, 198, 201, 208, Geometric Cone, 152, 160, 161, 168 210, 211 G-equivariant map, 231 Free Abelian topological group, 221 G-extension, 224 Free topological group, 221, 222, 227 G-homeomorphism, 218 Frolík’s class, 108, 109, 117, 119, 121, 123, G-homeomorphic embedding, 218 124, 126–129, 140, 141, 144, 206 Ginsburg’s question, 274, 278 Frolík sequence, 117, 119–121 G-invariant metric, 232 Frolík space, 19, 20 Glicksberg’s theorem, 24, 27, 28, 108, 130, Frolík theorem, 108, 117, 119, 123 225, 237 Full continuous extension, 259, 260 for p-bounded subsets, 139, 140 Function for bounded subsets, 108, 130–134, 136, b f -continuous, 124, 125 137, 139–142 br -continuous, 124 G-map, 218–220, 224, 231, 239 bounded, 59 G-orbit, 222 depends on ω coordinates, 32 GO-space, 170–173 γ f -continuous, 122 Group γr -continuous, 122 Abelian, 41 kr -continuous, 125 compact, 43, 48, 98, 104 left uniformly continuous, 58, 59, 145 completely metrizable, 182 ω-continuous, 210 complex numbers, of, 40 right uniformly continuous, 58, 145 connected, 104 sequentially continuous, 211 countably compact, 62, 72 uniformly continuous, 58, 59, 145 feebly compact, 40, 65, 66, 68, 69 z-closed, 133 first countable, 39, 47, 92 Function space, 108, 113, 122 integers, of, 40 294 Index

metrizable, 39, 47, 59, 60, 196 Hyperspace, 96 ω-cellular, 51 of nonempty closed subsets, 97 orientation-preserving homeomor- of nonempty compact subsets, 97, 98 phisms of the unit interval, of, 233 paratopological, 64 precompact, see totally bounded group I pseudocompact, 39, 44, 48, 50, 54, 55, I -almost disjoint sets, 263 59, 61–64, 66, 69, 71, 72, 78, 98, 101, Identity map, 130 103, 104, 111, 182, 220 Independent family, 281 quasitopological, 70 Inner automorphism, 42 quotient, 42, 48, 59 Interior, of a set, 65 rational numbers, of, 219 Invariant map, see G-map R-factorizable, 181 Invariant metric, 231, 232 second countable, 55, 69 Invariant set, 222, 223, 236, 244 semitopological, 64, 70 Irrational flow, 242 strongly pseudocompact, 104 Irrational numbers, 242 topological, 39 Isolated ordinal, 235 uY -discrete, 184 Group topology pseudocompact, 104 J strongly pseudocompact, 104 Jensen’s axiom, 215 G-saturation, of a subset, 222 G-space, 217–224, 228, 229, 231, 232, 236, 237, 241–245, 248 K k -space, 122 compact, 218, 223, 244 f k -space, 107, 122–125, 221, 222, 228, 237, countably compact, 219, 233, 235–237 r 240 G-normal, 241, 244–248, 250 k-space, 18–20, 60, 221 G-pseudocompact, 218, 241, 242, 247, Kernel, of a homomorphism, 42 248, 250 G-Tychonoff, 221–224, 228–231, 233, 236, 237 L locally compact, 233, 235 Layered family, 263 metrizable, 232 Left normal, 245 coset, 42 G-uniform function, 222, 223, 236, 237, group uniformity, 57 241, 244, 245, 247 induced uniformity, 57 bounded, 224, 242 translation, 64, 222 Level of an element of an η-layered family, 263 H Lightly compact space, 191 Hemicompact space, 222 Limit point, 6 Hereditarily maximal pseudocompact space, Lindelöf -group, 71 192, 201–205 Lindelöf property, 267 Hereditarily MP space, 192 Lindelöf space, 11, 16, 152, 154–158, 161, Hereditarily separable space, 49, 215 162, 166–168, 170, 171, 173, 174, Hewitt realcompactification, 109, 110, 115, 183, 255, 267–269 153, 218, 222 Linearly ordered , 112, 113 Homeomorphism, 112, 130, 218, 220, 234 initial and final segments, of, 112 Homogeneity, 41 left cofinal subset, of, 112, 113 Homogeneous space, 274 right cofinal subset, of, 112, 113 Homomorphism, 41, 42, 48 Linear order, 112 quotient, 54 Local π-base, 200 Hyperbounded subset, 115 Local π-character, 200 Index 295

Locally bounded space, 158 jointly continuous, 64 Locally compact separately continuous, 64 paratopological group, 73 μ-space, 121, 122, 141 point, at the left, 170 point, at the right, 170 space, 16, 122, 154, 155, 158–162, 165– N 169, 175, 195 Natural numbers, 17, 78, 108, 109, 117, 143, topological group, 40, 41, 60, 62, 72, 254 253 Locally convex algebra, 122 Natural projection, 2 Locally finite family, see essential family Natural uniformities, of a topological group, Locally pseudocompact space, 152, 158, 144–146 184, 221 Nested sequence, 174, 178 Locally pseudocompact topological group, Network, 212 221 n-Luzin gap, 259 Locally weakly pseudocompact Normal space, 8, 52, 112, 258 point, at the left, 170 point, at the right, 170 Luzin gap, 259, 265 O ω-, 210 ω-dense subset, 34, 36, 184 ω-monolithic space, 212 M ω-partition, 78, 84, 93 MA, see Martin’s Axiom One point κ-Lindelöf extension, 158 +¬ MA CH, 209, 210 Open cover, 143 MAD, see maximal almost disjoint family Orbit space, 222 Martin’s Axiom, 97, 259, 260 Oscillation of a function, 131, 137 Maximal almost disjoint family, 7, 195, 196, Oxtoby complete space, 175, 176, 182, 184– 198, 253–256, 258, 260–263, 265– 186 269, 272, 274–276, 278, 279 Maximal countably compact space, 192, 193, 210–215 P Maximal feebly compact space, 193–196, , 11, 111, 121, 125, 205 199 Paratopological group, 65 Maximal pseudocompact space, 192, 194, first countable, 64 196–203, 205, 206, 208–211, 214 locally compact, 73 Mazur property, 212, 213 second countable, 40, 69 Mazur space, 192, 212, 213 Partitioner set, 258, 266, 268 MCC-space, 192, 193, 209, 210, 214 p-bounded subset, 108, 109, 125, 126, 137, , 11 139–141 Metrizable hedgehog with κ spines, 161 Perfect mapping, 166 Metrizable space, 8, 11, 61, 111, 122, 152, Phase group, 217 155, 177, 181, 200, 213, 225, 231, Phase space, 218, 221, 224 232, 254, 257, 258 π-character, 200 Metrizable topological group, 39, 47, 59, 60, π-uniform, see G-uniform function 196, 231 p-limit, 81, 91, 125, 140, 142, 143 Monotone set, 282 Point-finite family, 10, 11 , 257, 258 Pointwise bounded family, 131 MP-space, 192, 209, 214 Pointwise convergence topology, 113, 122 Mrówka family, 266, 272, 286 Polish space, 181 Mrówka-Isbell space, 7, 253, 254, 257, 262, p-pseudocompact space, 77, 78, 81, 83, 85 263, 265, 274, 278, 280 Precompact group, 98, 100, 152 Mrówka MAD family, 266, 268, 269, 271, Preiss-Simon property, 192, 203, 204, 208, 272 209 Multiplication Preiss-Simon space, 192 296 Index

Pre-order Q Rudin, 83 Q-compactification, 219 Rudin-Keisler, 78 Q-point, 94 Projection map, 2, 144 Q-set, 257, 258 Property (a), 286 Quasitopological group, 70 Property (b), 108, 121–123, 125, 128, 129, Quotient 137 cone, 162 image, 221 Property (c), 129 map, 221 Pseudobase, 175, 176, 179 space, see orbit space Pseudocharacter, 45, 47, 48, 198 topology, 222 Pseudocompact extension, 199 group topology, 104, 196 R paratopological group, 73 Ra˘ıkov completion, 57, 58 quasitopological group, 70 Real z-ultrafilter, 15, 16 space, 1, 2, 6–11, 14–20, 22, 23, 28, 33, , 15, 16, 111, 121, 165, 34, 36, 39, 40, 44, 47, 52, 60, 61, 63, 222, 225 65, 79–82, 85, 87, 88, 90, 94, 97, 98, Real numbers, 109, 130, 131, 241 101, 104, 107–113, 115, 117–119, 121, Refinement, of a cover, 11 123–126, 128–130, 137, 142, 144, 146, Refining cover, 11 Y 152–156, 158, 159, 161, 167, 175, 183, Regular C<ω-discrete space, 183, 184 184, 186, 187, 192–195, 197, 199–201, Regular closed set, 8, 9, 225 203, 204, 206, 212, 214, 221, 222, 224, Relatively 227–232, 237, 240, 241, 254, 255, 274, countably compact subset, 275 275, 278, 279 dense subset, 242 topological group, 39, 44, 48, 50, 51, 54, pseudocompact space, 108, 194 55, 59, 61–64, 66, 69, 71, 72, 78, 98, Retract, 61 101, 103, 104, 109, 111, 140, 141, 182, R-factorizable group, 181 220, 229 Right Pseudocompactification of a space, 158 coset, 42, 53 Pseudocompactness group uniformity, 57 translation, 64 in k-spaces, 20 r-pseudocompact set, 108, 117, 121 of a product, 17, 19, 20, 28 strongly, 108, 121 (α, ) vs D -pseudocompactness, 85 R-quotient map, 213 vs Baire property, 10 Rudin-Keisler predecessor, 78 vs C-compactness, 34 Rudin-Keisler pre-order, 78 vs countable compactness, 6 Rudin pre-order, 83 vs Gδ-density, 34 vs dyadicity, 33 vs metacompactness, 11 S  vs ω-density, 34 -product, 11, 35  vs paracompactness, 11 δ-diagonal, 174  vs realcompactness, 15 <κ -product, 221 Sánchez-Okunev complete space, 175, 176, Pseudocompleteness, 175 181, 182, 184, 185 Pseudo-D-bounded space, 91 sb f -space, 122 Pseudo-intersection number, 275 Scattered space, 199, 201 Pseudometric, 145, 146 Second countable group, 40, 55, 69 Pseudonormal space, 8 Second countable space, 33, 34, 168 Pseudo-ω-bounded space, 89 Selection, 278 P-space, 25, 26, 121 Selective ultrafilter, 78, 83, 88 -space, 253 Semigroup, 109 Index 297

topological, 109 dyadic, 32–34, 36, 200, 213 Semiregularization, 66, 67 Eberlein compact, 204 Semitopological group, 64 Efimov, 39, 51 Separable space, 155, 163, 181, 211, 254, extremally disconnected, 225 255, 257, 258, 284 feebly compact, 40, 65, 66, 146, 191, Sequence 194, 196 centered, 175, 178 first countable, 92, 111, 122, 192, 195, complete, 175, 179 196, 202, 205, 208, 221, 225, 226, 228, nested, 174, 178 229, 254 Sequential closure, 262 F-pseudocompact, 99, 101 Sequentially closed subset, 262, 263 Fréchet-Urysohn, 197, 198, 201, 208, Sequentially compact space, 113 210, 211 Sequentially continuous function, 211 Frolík, 19, 20 Sequential order, 263–265 G-pseudocompact, 243 Set almost dominated by {n}, 280 hemibounded, 221, 222, 226, 227 Sets almost aligned, 280 hemicompact, 221, 222, 228 Simple compactification of a space, 164 hereditarily maximal pseudocompact, Simple pseudocompact space, 164, 165, 167, 192, 201–205 168, 177 hereditarily MP, 192 Slim of an η-layered family, 264 hereditarily separable, 49, 215 Sorgenfrey line, 64 homogeneous, 39, 41, 70 Souslin property, 49, 53 κ-metrizable, 61 Space, 2 κ-normal, 225 absolutely countable, 286 lightly, 191 accesible from a dense subset, 197, 198, Lindelöf, 11, 16, 152, 154–158, 161, 162, 205 166–168, 170, 171, 173, 174, 183, 222, almost compact, 16, 17 255, 268, 269 almost pseudo-ω-bounded, 19, 20 locally bounded, 123–125, 128, 129, 158 almost p-space, 176 locally compact, 16, 122, 154, 155, 158– (α, D)-pseudocompact, 77, 78, 85 162, 165–169, 175, 195, 221 arcwise connected, 184 locally pseudocompact, 152, 158, 184, Baire, 9–11 221 b f -space, 222, 226 maximal countably compact, 192, 193, Cech-complete,ˇ 174–176, 181 210–215 Y Cω -embedded, 183 maximal feebly compact, 193–196, 199 Cω-discrete, 183 maximal pseudocompact, 192, 194, 196– Y C<ω-discrete, 183–186 202, 205, 206, 208–211, 214 compact, 2, 6, 8, 11, 13–19, 26, 32, 34, Mazur, 192, 212, 213 35, 79, 80, 89, 97, 111, 192, 205, 208, metacompact, 11 220–222, 224, 231, 232, 240, 244, 246 metrizable, 8, 11, 61, 111, 122, 152, 155, compact metrizable, 32, 34–36, 90 177, 181, 200, 213, 221, 225, 231, 232, completely metrizable, 155, 175 254, 257, 258 completely regular, 222 Moscow, 225 connected, 94, 214, 278 normal, 8, 52, 112, 236, 245, 258 Corson compact, 204 of ordinals, 11 coset space, 221 of the rationals, 220 countable pseudocharacter, 225 ω-cellular, 49, 50 countably compact, 6–8, 15, 18, 35, 97, ω-monolithic, 212 111–113, 115, 143, 158, 176, 192, 201– Oxtoby complete, 175, 176, 182, 184– 203, 206, 208–214, 236, 254, 273 186 Dieudonné complete, 111, 112, 121, 225, paracompact, 11, 111, 121, 205 226, 228, 229, 231 p-pseudocompact, 77, 78, 81, 85 D-pseudocompact, 85, 86, 99, 101 Preiss-Simon, 192 298 Index

pseudo-χ(G)-compact, 229 invariant, 42, 46, 47, 59 pseudo-κ-compact, 229 torsion, 59 pseudocompact, see pseudocompact Subset space bounded, 158 pseudonormal, 8 C-compact, 34, 36, 109, 141–144 realcompact, 15, 16, 111, 121, 165, 222, C-embedded, 2, 4, 15, 16, 18, 26, 54, 63 225 C∗-embedded, 2, 13, 17, 24, 25 Y regular C<ω-discrete, 183, 184 cofinal, 178, 179, 181 relatively pseudocompact, 194 Gδ-dense, 14, 153, 184 Sánchez-Okunev complete, 175, 176, invariant, 42, 45 181, 182, 184, 185 nowhere dense, 65 second countable, 33, 34, 168 ω-dense, 34, 36, 184 separable, 155, 163, 181, 211, 254, 255, p-bounded, 109, 141 257, 258, 284 precompact, 111 σ-compact, 222, 232 regular closed, 61, 65 simple pseudocompact, 164, 165, 167, r-pseudocompact, 142 168, 177 sequentially closed, 262, 263 strongly accesible from a dense subset, strongly bounded, 108, 117–123, 125– 197–199, 209, 212, 213 127, 129, 140, 144 strongly p-pseudocompact, 89 strongly C-compact, 128, 129 strongly pseudocompact, 89, 101, 103 strongly relatively pseudocompact, 115 submaximal, 193–195, 199 strongly r-pseudocompact, 142, 144 submetrizable, 111, 112, 121, 141, 255 symmetric, 41, 56 suborderable, 279 with property (∗), 115 Talagrand compact, 204, 211 z-embedded, 55, 63 Todd complete, 175, 179, 184, 185 Supremum metric, 124 truly weakly pseudocompact, 155 Tychonoff, 222, 224 u-discrete, 183, 185 T ultrapseudocompact, 77, 89, 90, 92–94, Talagrand compact space, 204, 211 99, 101 Tamano theorem, 21, 23 universally maximal 3-cycle, 280 countably compact, 213, 214 Tietze-Urysohn extension theorem, 245 pseudocompact, 192–196 Tightness, 198, 214, 215 uY -discrete, 35, 36, 183, 184, 186 Tkachenko-Uspenskii’s theorem, 225 weakly pseudocompact, 152–163, 166– Todd complete space, 175, 179, 184, 185 171, 173, 174, 176, 177, 182–185, 187 Topological group, 98, 104, 108, 109, 219 zero-dimensional, 184, 254 almost metrizable, 60 Stabilizer subgroup, 228–231 compact, 219, 232, 243 Stationary subgroup, see stabilizer subgroup compact metrizable, 184 Stone-Cechˇ compactification, 13, 17, 28, 31, connected, 104, 230 33, 36, 78, 153, 218, 220, 224, 227, discrete, 220, 229, 230 237 Hausdorff, 217, 222 Strongly bounded subset, 108 hemibounded, 226–228 Strongly pseudocompact group, 104 Lie group, 231 Strongly pseudocompact group topology, Lindelöf, 219 104 locally compact, 40, 41, 60, 62, 72, 219, Strongly pseudocompact space, 77, 101, 103 223, 228, 229, 231, 241, 245, 247, 248, Subgroup, 42 250, 254 admissible, 45–49, 54, 108, 141, 144 locally precompact, 225, 226 C-compact, 109, 144 locally pseudocompact, 218, 220, 221, C-embedded, 54 228, 229, 231 closed, 62 metrizable, 220, 230–232 Index 299

Moscow, 225, 226 Universal maximal countable compactness, ω-narrow, 219, 220 192 Polish, 219 Universal weak selection, 281 precompact, see totally bounded group Urysohn’s lemma, 241, 242, 244, 245, 247 pseudocompact, 141, 144, 218, 221, 227, uY -discrete group, 184 229, 231, 232, 240 uY -discrete space, 35, 36, 183, 184, 186 second countable, 219 separable, 220 weakly pseudocompact, 152, 177, 182, V 184 Van Mill-Wattel selection problem, 278, 280 Topological group action, 122 Van Mill-Wattel theorem, 279 Topological groupcech Vietoris topology, 97, 273 Cech-complete,ˇ 181 Torus, 242 Tower number, 267 W Transformation group, see G-space Weak direct product, 221 Translation Weak P-point, 92, 93, 96 left, 64, 222 Weak pseudocompactness ( , ) right, 64 and C p X Y -spaces, 182 Trivial partitioner, 266, 268 and the Geometric Cone, 161 Truly weakly pseudocompact space, 155 and topological groups, 177 in terms of regular subrings of C∗(X), T3-, 65, 67 T.W.P., see truly weakly pseudocompact 186, 187 Tychonoff plank, 6 of a factor, 162 Type of ultrafilter, 78 vs Lindelöf property, 154 vs local compactness, 155 vs local pseudocompactness, 160 vs metrizable-like properties, 155 U vs Oxtoby completeness, 176 u-discrete space, 183, 185 Weakly orderable space, 278–280 Ultrafilter Weakly pseudocompact space, 152–163, free, 108, 109, 115, 125, 126, 137, 139– 166–171, 173, 174, 176, 177, 182– 143 185, 187 selective, 78, 83, 88 Weakly pseudocompact topological group, Ultrapseudocompact space, 77, 89, 90, 92– 152, 177, 182, 184 94, 99, 101 Weak selection, 278, 280 Uniform convergence, 233 Weight, 47, 192, 209, 210, 213, 267 on bounded subsets, 221 Weil completion, of a topological group, 43, Uniformity, 55, 57 44, 58, 228 compatible, 55, 57, 111, 146 Well embedded set, 115 finest, 146 WZ-mappings, 108 induced, 57 left, 57, 111, 144 natural, 146 Z right, 57, 111, 144 z-closed function, 21–26 two-sided, 57, 144 z-closed projection, 108, 122 Uniformly continuous function, 144–146 Zero-dimensional space, 184, 186, 199, 254 Uniform space, 111 Zero-set, 2, 13–16, 21–25, 30, 51, 52, 63, Unit circle group, 220 115, 116, 133, 152 Unit interval, 234–236, 247 ZFC, 258, 260, 263, 265, 267, 278 Universally maximal Zorn’s Lemma, 7, 33, 44, 193, 195, 255 countably compact space, 213, 214 z-ultrafilter, 13, 15, 16 pseudocompact space, 192–196 real, 15, 16