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A Absolutely Countably Compact Space, 286 Accumulation Point, 81 of a Sequence of Sets, 80 Action, 218, 220, 222–228, 231

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Index A B Absolutely countably compact space, 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 topology, 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 open set in a GO-space, 170, 173 Almost metrizable topological group, 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 -pseudocompact space, 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 Countably compact space, 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 normal space, 112 Cover, 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 topological space, 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
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