DOCSLIB.ORG

Author Index

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Author Index 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
Load more

Recommended publications
  • HISTORICAL NOTES Chapter 1:. the Idea of Topologizing the Set of Continuous Functions from One Topological Space Into Another To
    HISTORICAL NOTES Chapter 1:. The idea of topologizing the set of continuous functions from one topological space into another topological space arose from the notions of pointwise and uniform convergence of sequences of functions. Apparently the work of Ascoli [1883], [1889] and Hadamard [1898] marked the beginning of function space theory. The topology of pointwise convergence and the topology of uniform convergence are among the first function space topologies considered in the early years of general topology. The I supremum metric topology was studied in Frechet [19061. The paper of Tychonoff [1935] showed that the (Tychonoff) product on the set RX is nothing but the topology of pointwise convergence. In 1945, Fox [19451 defined the compact-open topology. Shortly thereafter, Arens [1946] studied this topology, which he called k-topology. Among other things which Arens proved was the compact-open topology version of Theorem 1.2.3. Set-open topologies in a more general setting were studied by Arens and Dugundji [1951] in connection with the concepts of admissible and proper topologies. Theorem 1.2.5 is due to Jackson [1952], and Example 1.2.7 can be found in Dugundji [1968]. Chapter 1:. Admissible [i.e., conjoining) topologies were introduced by Arens [1946] and splitting (i.e., proper) topologies were studied by Arens and Dugundji [1951], where they proved Theorem 2.5.3. Proofs of Theorem 2.5.2 and Corollary 2.5.4.a can be found in Fox [1945]. Corollary 2.5.7 is apparently due to Jackson [1952]; and Morita [1956] proved Corollary 2.5.8.
    [Show full text]
  • On a Special Metric
    Houston Journal of Mathematics c 2000 University of Houston ­ Volume 26, No. 4, 2000 ON A SPECIAL METRIC YUN ZIQIU AND HEIKKI J.K. JUNNILA Communicated by Jun-iti Nagata Abstract. In this note, we prove that whenever d is a compatible metric for a hedgehog space J having more than 2c spines, there exists ¯ > 0 and x J such that the family B (y; ¯): y B (x; ¯) contains more than 2 f d 2 d g c distinct sets. This result provides a negative answer to a question raised by Nagata in [6]. We also give positive answers to the same question under some extra conditions. 1. Introduction J. Nagata proved in [5] that each metrizable space X has a compatible metric d such that for every ¯ > 0, the family B (x; ¯): x X of all ¯-balls of (X; d) f d 2 g is closure-preserving, and in [6] he raised the following question: Does there exist, on each metrizable space X, a compatible metric d such that for every ¯ > 0, the family B (x; ¯): x X of all ¯-balls of (X; d) is hereditarily f d 2 g closure-preserving (as an unindexed family)? In this note, we answer the above question negatively by proving that there exists no compatible metric d on the hedgehog space H = J((2c)+) such that for every ¯ > 0, the family B (x; ¯): x H of all ¯-balls is point ¬nite or even f d 2 g point countable (as an unindexed family). We also show that the answer to the above question is positive under some added restrictions.
    [Show full text]
  • [Math.GN] 10 Apr 2002 O Vr Midset Every for N Scle a Called Is Rpriscnb Prahdtruhmidsets
    Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001 SPECIAL METRICS YASUNAO HATTORI Abstract. This is a survey on special metrics. We shall present some results and open questions on special metrics mainly appeared in the last 10 years. 1. Introduction This is a survey on special metrics. We shall present some results and open questions on special metrics mainly appeared in the last 10 years. We shall consider metrizable spaces and all metrics induce the original topology of a given metrizable space. For a metrizable space X, there are many metrics which induce the original topology of X. Some of them may determine topological properties of X. For example, it is well known that a metrizable space X is separable if and only if X admits a totally bounded metric, and X is compact if and only if X admits a complete totally bounded metric. 2. Properties related to midsets Let (X, ρ) be a metric space and y 6= z be distinct points of X. A set of the form M(y, z)= {x ∈ X : ρ(x,y)= ρ(x, z)} is called a midset or a bisector. Midsets are a geometrically intuitive concept, and several topological properties can be approached through midsets. arXiv:math/0204130v1 [math.GN] 10 Apr 2002 For example, the covering dimension of a separable metrizable space X can be characterized by midsets: Theorem 1 (Janos-Martin [11]). A separable metrizable space X has dim X ≤ n if and only if X admits a totally bounded metric ρ such that dim M ≤ n−1 for every midset M in X.
    [Show full text]
  • Vladimir V. Tkachuk Special Features of Function Spaces
    Problem Books in Mathematics Vladimir V. Tkachuk A Cp-Theory Problem Book Special Features of Function Spaces Problem Books in Mathematics Series Editors: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA For further volumes: http://www.springer.com/series/714 Vladimir V. Tkachuk ACp-Theory Problem Book Special Features of Function Spaces 123 Vladimir V. Tkachuk Departamento de Matematicas Universidad Autonoma Metropolitana-Iztapalapa San Rafael Atlixco, Mexico City, Mexico ISSN 0941-3502 ISBN 978-3-319-04746-1 ISBN 978-3-319-04747-8 (eBook) DOI 10.1007/978-3-319-04747-8 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014933677 Mathematics Subject Classification (2010): 54C35 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer.
    [Show full text]
  • Universal Free G-Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 157 (2010) 1296–1301 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Universal free G-spaces Natella Antonyan Departamento de Matemáticas, Division de Ingeniería y Arquitectura, Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Ciudad de México, 14380 México Distrito Federal, Mexico article info abstract Article history: For a compact Lie group G,weprovetheexistenceofauniversalG-space in the class of all Received 28 October 2008 paracompact (respectively, metrizable, and separable metrizable) free G-spaces. We show Received in revised form 11 March 2009 that such a universal free G-space cannot be compact. Accepted 27 March 2009 © 2009 Elsevier B.V. All rights reserved. MSC: 54H150 54B05 54D35 Keywords: Universal free G-space Join Orbit space Cross-section 1. Introduction By a G-space we mean a topological space X together with a fixed continuous action (g, x) → gx of a topological group G on X. In this paper we are mostly interested in free G-spaces, where the acting group G is compact Lie. Recall that a G-space X is free if for every x ∈ X the equality gx = x implies g = e,theunityofG.AG-space U is called universal for a given class of G-spaces G–P if U ∈ G–P and U containsasaG-subspace a G-homeomorphic copy of any G-space X from the class G–P. In [2] it is proved that if G is a compact Lie group then there exists a locally compact, noncompact free G-space which is universal in the class of all Tychonoff free G-spaces.
    [Show full text]
  • Equivariant Completions
    Comment.Math.Univ.Carolin. 35,3 (1994)539–547 539 Equivariant completions Michael Megrelishvili (Levy)* Abstract. An important consequence of a result of Katˇetov and Morita states that every metrizable space is contained in a complete metrizable space of the same dimension. We give an equivariant version of this fact in the case of a locally compact σ-compact acting group. Keywords: equivariant completion, factorization, dimension Classification: 54H15, 22A05 Introduction Let α : G×X → X be a continuous action of a topological group G on a uniform space (X, µ). We give a sufficient condition for the existence of a continuous extensionα ˆ : Gˆ × Xˆ → Xˆ where Gˆ is the sup-completion (i.e. the completion with respect to its two-sided uniformity) and (X,ˆ µˆ) is the completion of (X, µ). Our sufficient condition is necessary in the following important situation: Gˆ is Baire, µ is metrizable and for every g ∈ Gˆ the g-transition Xˆ → Xˆ isµ ˆ-uniformly continuous. As an application of a general equivariant completion theorem we unify the verification of the sup-completeness property for some natural groups. An important consequence of a result of Katˇetov [10] and Morita [14] states that every metrizable space is contained in a complete metrizable space of the same dimension (see Engelking [6, 7.4.17]). Using the G-factorization theorem [13] we obtain an equivariant generalization of the last fact in the case of locally compact σ-compact acting group G. This generalization, at the same time, improves some “equivariant results” of de Groot [7] and de Vries [16].
    [Show full text]
  • The $ K $-Spaces Property of Free Abelian Topological Groups Over
    THE k-SPACES PROPERTY OF FREE ABELIAN TOPOLOGICAL GROUPS OVER NON-METRIZABLE LASNEVˇ SPACES FUCAI LIN AND CHUAN LIU Abstract. Given a Tychonoff space X, let A(X) be the free Abelian topological group over X in the sense of Markov. For every n ∈ N, let An(X) denote the subspace of A(X) that consists of words of reduced length at most n with respect to the free basis X. In this paper, we show that A4(X) isa k-space if and only if A(X) is a k-space for the non-metrizable Laˇsnev space X, which gives a complementary for one result of K. Yamada’s. In addition, we also show that, under the assumption of ♭ = ω1, the subspace A3(X) is a k-space if and only if A(X) is a k-space for the non-metrizable Laˇsnev space X. However, under the assumption of ♭>ω1, we provide a non-metrizable Laˇsnev space X such that A3(X) is a k-space but A(X) is not a k-space. 1. Introduction In 1941, Markov [11] introduced the concept of the free Abelian topological group A(X) over a Tychonoff space X. Since then, the free Abelian topological groups have been a source of various examples and also an interesting topic of study in the theory of topological groups, see [2]. For every n ∈ N, let An(X) denote the subspace of A(X) that consists of words of reduced length at most n with respect to the free basis X. Recently, K. Yamada has showed the following two theorems: Theorem 1.1.
    [Show full text]
  • THE COMPACT-OPEN TOPOLOGY on C by Dissertation Submitted
    THE COMPACT-OPEN TOPOLOGY ON CCO by Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY i ii Mathematic·:. APPR01..,JED: R.A. McCoy, Chairman ·-· .J.T.Arnc•ld C. E, Au 11 P. Fletcher· ~i.R. Hcilub Blacksburg, Virginia "Malheur.!\ toi ·:.i tu di·:. que t•:•n bonheur· e:.t mor·t par·ce que tu n·'av.:tis ,, I pas reve pareil a eel a ton bonheur ... Le reve" de demain est une Joie. Mais la Joie de demain en est une autre; A / et rien heureusement ne re:.semble au reve qu·'on ·:. 1 en etait f.:tit car I c'est differemment que vaut chaque chose". A.Gide, "Les nourritures terrestres". THE COMPACT-OPEN TOPOLOGY ON CC:O by Ibul a Ntant•J Committee Chairman: Robert A. McCoy Mathematics. <ABSTRACT) This. paper· int,•estigates the- cc•mp.3.ct-open topology on the- s.e-t CKCX) of continuous real-valued functions defined on a Tychonoff space "'/\ I Mor·e pr·ecis.ely, 1,11e stud;.- the follo1,,.1ing probiem: If Pi::.; topological property, does there exist .:l. topological proi:•er·ty Q sc• that CK(X) has P if and only if X has Q ? Characteriz.ations. C•f many pr•:•pE'rtiE'·:.. ;r·e obtained tiH"•:•1Jghou1: the thesis, sometimes modulo some "mild" restrictions on the space X. The m~.in prc•per·ties in•.•olved ar·e s.1Jmm.:i.ri:ed in a di.:..gr·~.m in the introductic•n.
    [Show full text]
  • Absorbing Sets in Infinite-Dimensional Manifolds
    T. Banakh, T. Radul, M. Zarichnyi Absorbing sets in Infinite-Dimensional Manifolds Mathematical Studies Monograph Series Volume 1 VNTL Publishers Lviv, 1996 2 Taras Banakh, Lviv State University Taras Radul, Institute for Applied Problems of Mechanics and Mathematics Michael Zarichnyi, Lviv State University 3 Preface Many remarkable topological spaces can be characterized by means of short list of their topological properties. In early 70es the famous charac- terization theorems for Hilbert space manifolds and Hilbert cube manifolds were proved by Henryk Toru´nczyk.His technique was essentially based on the completeness of the considered spaces. A program of proving noncomplete counterparts of the Toru´nczyktheo- rems was realized in the fundamental paper of M.Bestvina and J.Mogilski. Starting from Anderson's cap-sets, Bessaga-Pe lczy´nski'sskeletoids, and ab- sorbing sets of J.West they cristallized the notion of absorbing set in Hilbert space manifold. These absorbing sets form an important class of spaces for which charac- terization theorems can be proved. Besides, they possess many remarkable topological properties, in particular, their topological and homotopy classi- fications coincide. The book is devoted to the theory of absorbing sets and its applications. It is divided into five chapters. The first chapter provides an exposition of the basic theory of absorbing sets. Our approach differs from that of Bestvina and Mogilski and is based on the first author's characterization of spaces that can be homotopy densely embedded into Hilbert space mani- folds. This allows us to simplify substantially our exposition and, moreover, to show the equivalence of the relative (Bestvina, Mogilski) and absolute (Dobrowolski, Mogilski) notions of absorbing sets.
    [Show full text]
  • Metrisable and Discrete Special Resolutions
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Topology and its Applications 122 (2002) 605–615 Metrisable and discrete special resolutions Kerry Richardson a,∗, Stephen Watson b,1 a Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand b Department of Mathematics and Statistics, 4700 Keele Street, York University, Toronto, Canada Received 10 November 2000; received in revised form 8 March 2001 Abstract In this article we give characterisations for a topological space constructed by special resolutions to be metrisable and discrete, respectively. We show how nice topological properties can be completely destroyed by resolutions. 2002 Elsevier Science B.V. All rights reserved. AMS classification: 54B99 Keywords: Special resolution; Fully closed maps 1. Introduction The method was introduced by Fedorcukˇ [3] and developed by Watson [10]. A lot of results of great importance have been obtained using this method. Among them; an example of a compact space X with different dimensions (covering dimension dimX = 2 and inductive dimensions satisfying 3 indxX 4foreveryx ∈ X) in [3], consistent examples of non-metrisable compact spaces with hereditarily separable finite powers in [7], and a CH example of a regular space X with a countable network and different dimensions (dimX = 1andindX = IndX 2) in [1]. \{ }→ Definition 1. Let X and Yx be non-empty topological spaces and fx : X x Yx be a continuous function for each x ∈ X. The point set is Z = {{x}×Yx : x ∈ X}. * Corresponding author. Partially supported by: The University of Auckland Postgraduate Research Fund; the Maori Education Trust; the New Zealand Mathematical Society; and a Marsden Fund Award, UOA611, from the Royal Society of New Zealand.
    [Show full text]
  • 1. Introduction Hedgehog Spaces Are an Interesting Source of Examples in Point-Set Topol- Ogy
    Pr´e-Publica¸c~oesdo Departamento de Matem´atica Universidade de Coimbra Preprint Number 21{26 FRAME PRESENTATIONS OF COMPACT HEDGEHOGS AND THEIR PROPERTIES IGOR ARRIETA, JAVIER GUTIERREZ´ GARC´IA AND JORGE PICADO Dedicated to Professor Bernhard Banaschewski on the occasion of his 95th birthday Abstract: This paper considers the compact hedgehog as a frame presented by generators and relations, based on the presentation of the frame of extended real numbers. The main focus will be on the point-free version of continuous and semi- continuous functions that arise from it, and their application in characterizations of variants of collectionwise normality. The variants to be considered are defined by selections of adequate families of sublocales and their characterizations depend on lattice-theoretic properties of the selected families. This way insertion and exten- sion results for semicontinuous and continuous functions with values in the compact hedgehog frame are generalized and unified. Keywords: Hedgehog space, compact hedgehog, frame, locale, zero sublocale, c z- and zκ-embedded sublocales, collectionwise normality, total collectionwise normality, compact hedgehog-valued frame homomorphism, insertion results, sublocale selection. 2020 Mathematics Subject Classification: 06D22, 18F70, 54D15. 1. Introduction Hedgehog spaces are an interesting source of examples in point-set topol- ogy. They constitute, in fact, three different classes of spaces [1]. Here we are interested in the so-called compact hedgehog spaces, from a point-free viewpoint. As the name suggests, the hedgehog can be described as a set of spines identified at a single point. Given a cardinal κ and a set I of cardinality S κ, the hedgehog with κ spines J(κ) is the disjoint union i2I R × fig of κ copies (the spines) of the extended real line identified at −∞: The compact hedgehog space ΛJ(κ) is the hedgehog endowed with the (com- pact) topology generated by the subbasis f (r; |)i j r 2 Q; i 2 I g [ f (|; r)i j r 2 Q; i 2 I g where (r; |)i := (r; +1] × fig and (|; r)i := J(κ) n ([r; +1] × fig).
    [Show full text]
  • Strongly Base-Paracompact Spaces
    Comment.Math.Univ.Carolin. 44,2 (2003)307–314 307 Strongly base-paracompact spaces John E. Porter Abstract. A space X is said to be strongly base-paracompact if there is a basis B for X with |B| = w(X) such that every open cover of X has a star-finite open refinement by members of B. Strongly paracompact spaces which are strongly base-paracompact are studied. Strongly base-paracompact spaces are shown have a family of functions F with cardinality equal to the weight such that every open cover has a locally finite partition of unity subordinated to it from F. Keywords: base-paracompact, strongly base-paracompact, partition of unity, Lindel¨of spaces Classification: 54D20 1. Introduction The weight, w(X), of a topological space X is the minimal cardinality of a basis for X. A space X is base-paracompact if there is a basis B for X with |B| = w(X) such that every open cover of X has a locally finite refinement by members of B. The author studied these spaces in [Po] as a weakening of totally paracompact spaces. Is strong paracompactness witnessed by a base in the same manner as base-paracompact spaces? A family A of subsets of a topological space X is said to be star-finite if for every A ∈ A the set {B ∈ A : B ∩ A 6= ∅} is finite. A topological space X is strongly paracompact if every open cover of X has a star-finite open refinement. A topological space X is strongly base-paracompact if there is a basis B for X with |B| = w(X) such that every open cover of X has a star-finite open refinement by members of B.
    [Show full text]