Author Index

Aleksandrov, P.S. 36,45,64,87,88, 134, 136, Hausdorff, F. 30, 87, 89, 90 140, 147, 156, 163, 168, 169, 177, 185 Henderson, D.W. 163 Alexandroff, P.S. (= Aleksandrov, P.S.) Hilbert, D. 96, 99 Alexandrov, P.S. (= Aleksandrov, P.S.) Hopf, H. 45,88, 168 Anderson, R.D. 40, 188 Hurewicz, W. 89,96, 147, 160, 169, 179, 182 Arhangel'skii, A.V. (= ArkhangeJ'skii) Arkhangel'skii A.V. 4,26,40,48,87,88, 160, Isbell, J.R. 89 189 Ivanov, A.V. 135, 147

Baire, R. 21,88 Jensen, R.B. 145 Banach, S. 55 Jordan, M.E.C. 99, 136 Bockstein, M.F. (= Bokshtein M.F.) Junnila, H.J. 38 Bokshtein, M.F. 96,173, 178 Boltyanskii, V.G. 96, 176 Katetov, M. 96,148,150,152,153 Borel, E. 57 Keldysh, L.V. 40,89,97, 185, 188 Borsuk, K. 96, 136 Kelley, lL. 89 Bourbaki, N. 88 Knaster, B. 111 Brouwer, L.E.J. 67,89,95, 101, 106, 120 Kodama, Y. 96 Kolmogorov, A.N. 97,185 Cantor, G. 95, 101, 109 Kowalsky, H.J. 21,89 Cech, E. 63, 89, 96, 154 Kozlovskii, I.M. 186 Cohen, D.E. 64, 89 Krein, M.G. 57 Kunneth, D. 167 Descartes, R. 98 Kuratowski, K. 96, 111, 147 Dieudonne, J.A. 85 Kurosh, A.G. 75 Douwen, E.K. van 155 Kuz'minov, V.I. 96 Dowker, C.H. 96, 148, 149, 154, 155, 177 Dranishnikov, A.I. 97,170 Lashnev, N.S. 38, 89 Dugundji, J. 144 Lavrent'ev, M.A. 53,89 Dyer, E. 96 Lebesgue, H.L. 56,90, 106 Levshenko, B.T. 97,158,162,163,190 Engelking, R. 87, 89 Lifanov, I.K. 143, 146 Euclid 98 Lokutsievskii, a.v. 96, 141, 156 Lunts, A.L. 96, 141 Fedorchuk, V.V. 4, 89, 97, 142, 144, 145, 147, Luxemburg, L.A. 97, 158 156,159,163,183,184,188,190 Fermat, P. 98 Mardeshich, S. 138 Filippov, V.V. 89,97, 143, 145, 155, 183 Martin, J. 145 Franklin, S.P. 26,89 Mazurkiewicz, S. 100, 135 Fraudenthal, H. 138 Menger, K. 95, 102, 113, 135, 155 Frechet, M.R. 48, 89 Michael, E. 38,89 Milman, D.P. 57 Giihler, W. 89 Moebius, A.F. 174 Morita, K. 96, 148, 152, 180, 182 Hahn, H. 100 Mysior, A. 89 194 Author Index

Nagami, K. 148,152,189 Sklyarenko, E.G. 96,140,163,164,183, 189 Nagata, 1. 89,153,161,189 Smirnov, Yu.M. 96,155,158,160,161,164, Naimark, M.A. 89 189 Nedev, S. 89 Sperner, E.C. 119 Nemets, A.G. 153 Stoilow, S 40,90 Nemytskii, V.V. 29 Stone, A.N. 22,90 Niibeling, G. 36,55,96,102,126 Stone, M.H. 57, 63, 80, 90 Noble, N. 61,89 Novak,1. 89 Terpe, F. 90 Tietze, H. 66 Oxtoby, 1.c. 89 Tikhonov, A.N. 31,36,50,63,69,75,80,86,90 Tukey,l.W. 90 Pasynkov, B.A. 88,89,90,96,97,139,140,143, Tumarkin, L.A. 96,147,152,164 146,152,159,161,165,183,184,185,187,192 Peano, G. 90,99 Urysohn, P.S. (= Uryson P.S.) Pears, A.R. 158 Uryson, P.S. 31,48,50,66,87,88,90,95,102, Poincare, H. 55,90, 106 131, 147, 172 Pol, E. 155 Uspenskii, V.V. 90 Pol, R. 97,155,163,190 . Ponomarev, V.L 38, 39, 87, 89 van Douwen, E.K. 155 Pontryagin, L.S. 36,55,89,96,102,126,174, Vedenisov, N.B. 140 176,186 Vopenka, P. 150 Prat,l.1. 155 Przymusinski, T. 155 Walker, R.C. 90 Wallman, H. 89, 160 Roy, P. 145,151,155 Walsh,1.1. 164 Rudin, M.E. 90 Watson, W.S. 90 Rudin, W. 90 Weierstrass, K.T.W. 57, 80 Weiss, W.A. 145 Savinov, N.V. 184 Weil, A. 90 Shchepin, E.V. 81,90,156 Whyburn, G.T. 40,90 Shtanko, M.A. 134 Shvedov, LA. 155 Zarelya, A.V. 96,140,151,164,182 Sierpinski, W. 101, 135 Zariski, O. 27 Sitnikov, K.A. 96,125,135,153,172,178 Zolotarev, V.P. 145 Subject Index

A-decomposition 41 - space of countable weight 70 A-essential map 162 - space B(,) 37 A-strongly infinite dimensional 162 - space of weight, 21 A-weakly infinite dimensional 162 BalIs 20 Absolute extensors 144 Barycentric coordinate system 114, 123 Absolutely kappa-normal space 144 - coordinates 114 Abstract closure operator 9 - map 124 - nearness relation 9 - -, generalized 124 - praclosure operator 25 -- with respect to a partition of unity 124 - simplex 116 - subdivision 117 - simplicial complex 116, 148 Base 6 Addition sequence of a triad 168 - of a space at a point 14 - theorem, Dowker's 154 - of a space at a set 14 Additivity property of a network 18 -, pointwise countable 39 Adherent colIection of subsets 11 -, wealthy 178 - point 11 Basis of a uniformity 82 Admissible variation of continuous map 162 Body of a complex 116 Affinely independent 114 Bokshtein coboundary operator 173 Alexandrov supersequence 15 - inequalities 173, 178 Alekandrov's axiom system 156 Boundary 16 - general problem 169 Bounded metric 21 - identity 168 Brouwer axiom 156 - imbedding theorem 36 Brouwer's fixed point theorem 120 - obstruction theorem 170 - theorem on invariance of the dimension of - problem 141,169 Euclidean space 120 Aleksandrov-Borsuk theorem 136 Brouwer-Tietze-Uryson extension theorem 67 Aleksandrov-Cech homology 167 Antidiscrete topology 5 Canonical map 123 Associated separated uniformity 83 -- theorem 123 ---lemma 83 - quotient map associated to an intersection of Attaching map 45 topologies 49 - one space to another 45 - topology of a decomposition 41 Axiom of choice 69 Cantor cube 70 - of countability 14 - curve 101 -, Hausdorff 28 - discontinuum 70, 80, 109 - To 27 - discontinuum of countable weight 70

-Tj 27 - manifold 134, 147

-T2 27 - -, infinite dimensional 165 -T3 2&, - -, of dimension n 134 Axioms for dimension functions 155 - perfect set 10,34, 102, 108, 133 -, separation 27 - staircase 109 Cartesian product of sets 67 Baire category theorem 54 Category, smalI 73 - property 53, 54 Cauchy filter 84 - space B(~o) 34 - sequence 52 196 Subject Index

Cech's monotonicity theorem 154 - spectrum 79 - sum theorem 154 -, sequentially 32 Center of a simplex 117 Continuum 72 Centered collection of subsets 11 - Hypothesis 143 Chain of sets 11, 71 -, one-dimensional 102 Close subsets (with respect to a proximity) 86 Contraction mapping 55 Closed cover 30 Convergence, uniform 21 - map 37 - of a (centered) collection of sets 11 - k-dimensional simple 114 Convergent collection of open sets 164 - r-dimensional face 115 - sequence 12 - set 8 Countable dimensional 160 - subcomplex 116 - Frechet-Uryson fan 15 Closure operator, abstract 9 - multiplicity 185 - operator 8 --, map of 39 - 8 - sum axiom 155 Cofinal subspectrum 77 - to one map 39 Cohomological dimension 167, 177 - pseudocharacter, spaces of 14 -- of paracompact spaces 177 - weight, Hilbert space of 20 Cohomology groups with compact - compact 58 support 166 Cover 30 Combinatorial shrinking lemma 30 -, closed 30 - star of a simplex 116 -,open 30 - thickening lemma 31, 106 -, shrinking of 30 Compact map 65 Curve 102 - metric space 60 - set of weight, 80 Decomposition, completely continuous 45 - (topological space) 57 -, lower continuous 44 Compactification 62 -, upper continuous 44 - axiom 155 - space 41 -, Hausdorff 62 - of a topological space 41 -, Stone-Cech 63 Dense subdivision between disjoint sets 66 Compactum 62 Dg 106 Comparing classes of topological spaces, Diagonal product 68 problem of 36 Diameter (of a set) 21 Comparison of sets 31 Dieudonne complete 85 Compatible collection of topologies 49 dim 106 Complete complex 115 Dimension of a complex 116 - metric 52 - of an abstract simplex 116 - uniform space 84 - Dg 106 Completely continuous partition 45 - ofa map 179 - paracompact spaces 144 Dimensional component 147 - regular 50 Dimensionally full-bodied 176 Completion of a metric space 56 Dimensiongrad 106 Complex 115 Directed 73 Component, connected 72 Direction (in a set) 79 Composition of morphisms 76 Disconnected 71 Condensation 33,36,51, 109 Discontinuous 31 CQnnected 71 Discrete map 40 - subspace 71 - space 5 - component 72 - topology 5 - two point set topology 23 Distance zero from a set 7 Continuous 31 - between points 20 - map of a metric space 32 - between two subsets 20 - map of a topological space 32 - from a point to a set 20 Subject Index 197

- from a point to a set (in a prametric space) 23 -- (of topological spaces) 10 - between points (in a prametric space) 23 Full-bodied 291 Distant subsets (with respect to a proximity) 86 Fully closed map 184 Distinguished subsets (with respect to a Functional separation 50 partition) 41 Functionally Hausdorff 50 Double restriction 47 Functions of first Baire class 12, 13 Dowker addition theorem 154 -- second Baire class 13 Dowker's recursion theorem 154 Fundamental sequence 52 - theorem 149 - theorem of algebra 39 Dugundji spaces 144 Dyadic compacta 144 y-distinguished set 47 Dyadic compacta of class AE(O) 144 y-distinguished subset 41 General position 114 Eilenberg-Maclane complex 169 - topology 4 Elements of inverse system 73 Generalized barycentric map 124 Enlargement of a cover 107 - Hilbert space of weight T 21 e-tJ definition of continuity 31 - Hilbert space 21 e-shift 122 - Jordan theorem 136 e-dense subset 51 - Menger conjecture 155 e-discrete subset 51 Ga-set 53 e-map 122 Geometrical realization ofa triangulation 118 Equivalent (metrics) 21 - simplicial complex 116 - Cauchy sequences 56 Geometrically independent 114 Essential 121 - map to Hilbert cube 162 Hahn-Mazurkiewicz theorem 101 -- theorem 127 Hausdorff compactification 62 Euclidean metric 20 -, functionally 50 Everywhere dense subset 15 - space 28 Exact morphisms 77 -axiom 28 Extension of a topological space 85 Hedgehog space 21 Hedgehog space of spininess T 21 Face of an abstract simplex 116 Hereditarily monotone topological Factorization of a map in a spectrum 81 invariant 154 -lemma 80 - monotone with respect to open subsets -lemma for mappings 81 154 Fan product 68 - quotient map 48 -- of spaces 69 - satisfies the sum theorem 154 -- of maps 69 - uncountable dimensional 163 Far 7 Hilbert cube 70 Fiber product 68 - space 20 Filter 84 Hilbert space of countable weight 20 Fine uniformity 83 - -, (generalized) of weight T 21 Finite multiplicity, map of 39 Homeomorphic maps 80 - sum axiom 156 - topological spaces 33 - to one map 39 Homeomorphism 33 First axiom of countability 14 Homological dimension 167 - Baire class, functions of 12 Hopf's Theorem 168 - Baire class, functions of 13 Hurewicz homomorphism 169 - category set 54 HcS" 166 - "parallel" lemma 71 Fixed point theorem 121 Imbedding 36 Frechet-Uryson fan 15 - problems 36 - space 13,41,48 Ind 104 Free sum (of topologies) 10 ind 104 198 Subject Index

Independent, affinely 114 - subsets 15 -, geometrically 114 - weight 157 Induced topology 9 Lebesgue dimension dim 106 - uniformity 82 Lexicographic ordering on the square 19 Inductive dimension, large 104 Light map 40 - dimension 104 Limit of maps 77 - dimension, small 104 - of an inverse system 74 - kernel 147 - - a collection of open sets 164 Inductively zero-dimensional 50,51 Limiting element of a partially ordered set 77 - zero-dimensional space 37 - projections of an inverse system 74 Infinite dimensional Cantor manifold 165 Linearly ordered space 18 - dimensional 107 Local homeomorphism 40, 185 Integrality axiom 155 Locally compact 63 Interior product 68 - connected 72 - of a set 17 - path connected 72 Internal dimensional kernel 147 Lower continuous partition 44 Intersection of metrizable topologies 25 - of topologies 5 Maier-Vietoris exact sequence 168 -- -, canonical quotient map associated to 49 Main "internaF'tasks of general topology 4 Interval 10 Map of countable multiplicity 39 of a linearly ordered set 18 - of finite multiplicity 39 Inv 76 - quotient 43 Inverse sequence 51 -, attaching 45 - spectrum 73 -, closed 37 - system 73 -, compact 65 -- of sets 73 -, countable to one 39 -- of topological spaces 73 -, discrete 40 - -, limit of 74 -, finite to one 39 - -, elements of 73 -, fully closed 184 - -, limiting projections of 74 -, hereditarily quotient 48 - -, projections of 73 -, light 40 - -, thread of 75 -, monotone 40

InvA 77 -, open 37 Isolated point 5 -, parallel to a space 184 Isometry of a metric space 56 -, perfect 66 Isomorphism of complexes 118 -, proper 65 - of inverse systems 77 -, restriction of 46 -, totally zero-dimensional 180 Jensen's combinatorial principle 145, 183 -, uniformly continuous 82 Join of spaces 45 -, zero-dimensional 180 Jordan curve 99 Maps diagonal product of 68 -, fan product of 69 k-covering map 66 -, fiber product of 69 k-map 65 -, homeomorphic 80 k-space 64 -, interior product of 68 kR-space 65 -, limit of 77 Knaster-Kuratowski fan III -, product of 68 Kowalsky's theorem 21 -, union of 40 Kunneth formula 166 Mardeshich's Factorization Theorem 137 Kuratowski-Zorn lemma 146 Martin's axiom 145, 183 Kurosh's theorem 75 Maximal connected set 72 Menger's axiom system 155 Large base 157 - theorem 135 - inductive dimension 104 Metric 20 Subject Index 199

- dimension 153 - map 37 - proximity 86 - sets 5 - space 20 - star 117 - space, compact 60 Open-closed subset 17 - -, complete 52 Openly-generated compactum 144 - uniformity 82 - k-dimensional simplex 114 -, bounded 21 - r-dimensional face 115 -, equivalent 21 - subcomplex 116 -, Euclidean 20 -, stronger 22 P-space 32 -, trivial 20 Parallel to a space 184 -, totally bounded 52 Parallels lemma 71 -, usual on IR 20 Partial pre-order 73 Metrizability, by a totally bounded metric 52 Partition 103, 129 Metrizable topological space 22 -, thin 104 - topology, intersections of 25 Path connected 72 Metrization, by a complete metric 53 --, local 72 Moebius band modulo p 174 Peano curve 99 Monotone map 40 - continuum 100 Monotonicityaxiom 155 - umbrella 99 - of dim with respect to closed sets 108 Perfect map 66 - of dimension 144 -set 109 - of Ind with respect to closed subsets 105 Perfectly kappa-normal space 144 -, of ind 105 Plane, Nemytskii 29 - theorem, Cech's 154 Poincare axiom 156 Morphism of an inverse system 75 - duality isomorphism 171 Mushroom lemma 127 Pointwise convergence 12 - countable base 39 Natural topology on the plane 6 Polyhedron 116 - topology on IR 6 Pontryagin duality 167 -- on the partition 41 Power of a space 68 Near 7 Praclosure 24 Nearness relation 7 - operator 24 -- induced by a topology 7 - -, abstract 25 Neighbourhood 7 -- generated by a nearness relation 25 Neighbourhoods, spherical 20 Prametric 23 Nemytskii plane 29 - space 23 Nerve 116 -, strong 24 Nested interval principle 61 -, symmetric 23 Network 17 -, separating 23 -, additivity property of 18 Precompact 85 Niibeling-Pontryagin imbedding theorem 36 Prime spectrum of a ring 28 - theorem 126 Principal star 117 Normal cover 83 Product of maps 68 - space 29 -- topological spaces 67 - -, nonmetrizable 31 -- uniformities 84 Normalization axiom 155 -, diagonal 68 Nowher~ dense subset 16 -, fiber 68 Projection map (to partition space) 43 Obstruction 170 Projections of fan product 69 Omega-discrete 180 -- an inverse system 73 --map 122 Projective system of sets 73 One-dimensional continuum 102 system 73 Open cover 30, 57 Proper map 65 200 Subject Index

Proximity 86 Sigma-complete 80 - generated by a uniformity 86 Sigma-spectrum 81 - space 86 Simplex 115 -, larger 86 Simplicial complex, abstract 116 -, metric 86 - -, geometrical 115 -, topology induced by 86 - map 118 -, uniform 86 Sitnikov's example 135 Pseudocharacter, countable 14 Size 114 Pseudocompact 61 Skeleton 115, 116 Pseudointerior 162 Small category 73 Pseudometric 83 - inductive dimension 104 Pseudometric subordinate to a uniform - inductive dimension at a point 104 cover 83 - subsets 15 Pseudo-open 48 Solenoid 51 Sorgenfrey line 4, 14, 16, 19,22,34,37 Quotient map 43 Sp 76 - -, hereditarily 48 SPA 77 - topology 42 Space 14 -, prametric- 2-3 ~-compactum 70 -, Tikhonov 50 Real compact space 70 Spaces of countable pseudocharacter 14 Recursion theorem, Dowker's 154 Spectra 74, 77 Refinement 82 Spectral cohomology 178 Regular space 28 - decomposition of metrizable compacta 138 Relative cohomological dimension 177 - theorem for mappings 81 - dimension 130 - topology 27 Restriction of a map 46 Spectrum, continuous 77 --- -, double 47 -, of a ring 28 -, over a direction 79 s-map 39 -, sigma-complete 80 S-strongly infinite dimensional 162 -, .-complete 80 S-weakly infinite dimensional 162 Sperner's lemma 119 Second "parallel" lemma 71 Spherical neighbourhoods 20 - Baire class, functions of 13 Spininess 21 - category set 54 Star refinement 82 Semicontinuum 135 Steenrod-Sitnikov homology 178 Separable topological space 16 Stone-tech compactification 63 Separated uniformity 83 Stone-Weierstrass theorem 80 -- associated to a uniformity 230 Strong prametric 24 Separating prametric 23 Strongly infinite dimensional 161 Separation axioms 27 - paracompact space 130 - -, functional 50 Subbase 5 -, of a space by a subset 134 Subcomplex 116 Sequential closure 13 Subcover 57 -- operator 13 Subdivision 117 - topological space 12 Subordinate pseudometric lemma 83 Sequentially compact 57 Subspace 9 - continuous 32 Sum theorem for dim 129 Servant subgroup 173 --- zero-dimensional sets 112 Shchepin theorem for homeomorphisms 81 --,tech's 154 Shift of size less than 114 Supersequence, A1exandrov 15 Shrinking (of a cover) 30 Suslin continuum 144 Sierpiilski carpet 101, 133 Symmetric 23 Sierpiilski's theorem 135 - prametric 23 Subject Index 201

Symmetry axiom 20 Totally bounded 52 -- uniform space 85 To axiom 27 - disconnected 110

T1 axiom 27 - discontinuous 110 T2 axiom 27 - zero-dimensional map 180 T3 axiom 28 1;-space 31 !-complete 80 Transfinite dimension ind 157 Thickening lemma 106 Triangle axiom 20 Thin partition 104 Triangulation in a Hilbert space 148 Thread of an inverse system 75 - of a polyhedron 116 Tikhonov cube 70, 80 - 115 Tikhonov product 67 Trivial metric 20 - space 50,51 Tikhonov's imbedding theorem 36 Uncountable dimensional 163 Tikhonov's lemma 69 Underlying polyhedron of a complex Top 69 116 Topological invariance axiom 155 Uniform convergence 21 - invariant 4, 33 - proximity 86 - product 67 - space 82 - property 33 - structure 82 - space 5 - topology 82 -, Dieudonne complete 85 - space, precompact 85 - -, inductively zero-dimensional 50 - -, totally bounded 85 - -, normal 29 Uniformities, product of 84 topological space, partition of 41 Uniformity 82 - -, regular 28 - compatible with a topology 82 - -, separable 16 -, basis of 82 - -, sequential 12 -, fine 83 - -, uniformly complete 85 -, induced 82 Topological spaces, comparison of 31 -, metric 82 - -, fan product of 69 -, universal 83 - -, inverse system of 73 Uniformly complete 84 - -, product of 67 - continuous map 82 Topology 5 Union of maps 40 - associated to a uniformity 82 Universal coefficient formula 166 -, compact 57 - Menger compactum 133 -, completely regular 50 -- curve 103, 133 - generated by a linear ordering 18 - weakly countable dimensional spaces 161 -- by a metric 20 - object problem 36 -- by a pradosure operator 26 - space for spaces with a countable base 20 -- by a prametric 23 - uniformity 83 -- by a topological space 9 Upper continuous partition 44 -, Hausdorff 28 Uryson's lemma 50,66 - induced by a proximity 86 - general problem 172 - of pointwise convergence 12 Uryson's problem 172 -, locally compact 63 - theorem 131 -, quotient 42 Uryson-Menger formulas 113 -, spectral 27 Usual metric on ~ 20 -, stronger 5 - topology on the plane 6 -, uniform 82 --on ~ 6 -, usual on R" 20 --on~" 20 -, weaker 5 -, Zariski 27 Weak topology generated by a collection Topy 69 of maps 51 202 Subject Index

Weakly countable dimensional 161 Weiss's theorem 145, 183 - infinite dimensional 162 Wealthy base 178 Zariski topology 27 Weight of a map 80 Zero-countable dimensional 160 -, of Baire space 21 Zero-dimensional map 180 Encyclopaedia ofMathematical Sciences Editor-in-chief: R. V. Gamkrelidze

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