DOCSLIB.ORG

Vladimir V. Tkachuk Special Features of Function Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface This is the second volume of the series of books of problems in Cp-theory entitled A Cp-Theory Problem Book, i.e., this book is a continuation of the first volume subtitled Topological and Function Spaces. The series was conceived as an introduction to Cp-theory with the hope that each volume will also be used as a reference guide for specialists. The first volume provides a self-contained introduction to general topology and Cp-theory and contains some highly nontrivial state-of-the-art results. For example, Sect. 1.4 presents Shapirovsky’s theorem on the existence of a point-countable - base in any compact space of countable tightness and Sect. 1.5 brings the reader to the frontier of the modern knowledge about realcompactness in the context of function spaces. This present volume introduces quite a few topics from scratch but dealing with topology and Cp-theory is already a professional endeavour. The objective is to study the behaviour of general topological properties in function spaces and establish the results on duality of cardinal functions and classes with respect to the Cp-functor. The respective background includes a considerable amount of top- notch results both in topology and set theory; the author’s obsession with keeping this work self-contained implied that an introduction to advanced set theory had to be provided in Sect. 1.1. The methods developed in this section made it possible to present a very difficult example of Todorceviˇ cˇ of a compact strong S-space. Of course, it was impossible to omit the famous Baturov’s theorem on coinci- dence of the Lindelöf number and extent in subspaces of Cp.X/ for any Lindelöf ˙-space X and the result of Christensen on -compactness of X provided that Cp.X/ is analytic. The self-containment policy of the author made it obligatory for him to give a thorough introduction to Lindelöf ˙-spaces in Sect. 1.3 andtothe descriptive set theory in Sect. 1.4. We use all topological methods developed in the first volume, so we refer to its problems and solutions when necessary. Of course, the author did his best to keep every solution as independent as possible, so a short argument could be repeated several times in different places. v vi Preface The author wants to emphasize that if a postgraduate student mastered the material of the first volume, it will be more than sufficient to understand every problem and solution of this book. However, for a concrete topic, much less might be needed. Finally, let me outline some points which show the potential usefulness of the present work: • The only background needed is some knowledge of set theory and real numbers; any reasonable course in calculus covers everything needed to understand this book. • The student can learn all of general topology required without recurring to any textbook or papers; the amount of general topology is strictly minimal and is presented in such a way that the student works with the spaces Cp.X/ from the very beginning. • What is said in the previous paragraph is true as well if a mathematician working outside of topology (e.g., in functional analysis) wants to use results or methods of Cp-theory; he (or she) will find them easily in a concentrated form or with full proofs if there is such a need. • The material we present here is up to date and brings the reader to the frontier of knowledge in a reasonable number of important areas of Cp-theory. • This book seems to be the first self-contained introduction to Cp-theory. Although there is an excellent textbook written by Arhangel’skii (1992a), it heavily depends on the reader’s good knowledge of general topology. Mexico City, Mexico Vladimir V. Tkachuk Contents 1 Duality Theorems and Properties of Function Spaces ................... 1 1.1 Some Additional Axioms and Hereditary Properties.................. 2 1.2 Monolithity, Stability and Their Generalizations ...................... 11 1.3 Whyburn Spaces, Calibers and Lindelöf ˙-Property ................. 19 1.4 A Glimpse of Descriptive Set Theory .................................. 27 1.5 Additivity of Properties: Mappings Between Function Spaces ....... 35 1.6 Bibliographic Notes ..................................................... 42 2 Solutions of Problems 001–500 ............................................. 45 3 Bonus Results: Some Hidden Statements ................................. 527 3.1 Standard Spaces ......................................................... 528 3.2 Metrizable Spaces ....................................................... 529 3.3 Compact Spaces and Their Generalizations ........................... 530 3.4 Properties of Continuous Maps......................................... 531 3.5 Completeness and Convergence Properties ............................ 532 3.6 Product Spaces .......................................................... 533 3.7 Cardinal Invariants and Set Theory .................................... 535 3.8 Raznoie (Unclassified Results) ......................................... 536 4 Open Problems ............................................................... 539 4.1 Analyticity and Similar Properties ..................................... 540 4.2 Whyburn Property in Function Spaces ................................. 541 4.3 Uniformly Dense Subspaces ........................................... 542 4.4 Countable Spread and Similar Properties .............................. 544 4.5 Metacompactness and Its Derivatives .................................. 547 4.6 Mappings Which Involve Cp-Spaces .................................. 551 4.7 Additivity of Topological Properties ................................... 552 4.8 Raznoie (Unclassified Questions) ...................................... 554 vii viii Contents Bibliography ...................................................................... 557 List of Special Symbols .......................................................... 573 Index ............................................................................... 577 Detailed Summary of Exercises 1.1. Tampering with Additional Axioms: Some Hereditary Properties Equivalent conditions for hl.X/ Ä . Problem 001. Characterizing hereditarily/perfectly normal spaces . Problems 002–003. Right-separated and left-separated spaces . Problems 004–009. Invariants s; hl; hd in finite and countable powers . Problems 010–013. Spread of X X is an upper bound for minfhd.X/; hl.X/g . Problem 014. A bound on cardinality of X using hl.X/ . .Problem 015. .X X/ Ä '.Cp.X// if ' 2fs; hl; hdg and is its dual . .Problems 016–018. Hedgehogs and duality theorems for s;hl;hd . Problems 019–035. Properties s Ä ! and hl Ä ! can be multiplicative in Cp.X/ . Problem 036. Non-separable spaces of countable spread . Problem 037. The Delta-lemma . Problem 038. CH constructions for distinguishing s; hl and hd. Problems 039–040. Compact spaces and their points of Ä ! under CH . Problem 041. There are P -points in ˇ!n! under CH
Load more

Recommended publications
  • Arxiv:Math/0412498V1
    A semifilter approach to selection principles Lubomyr Zdomsky February 8, 2020 Abstract In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal g is a lower bound of the additivity number of the σ-ideal generated by Menger subspaces of the Baire space, and under u < g every subset X of the real line with the property Split(Λ, Λ) is Hurewicz, and thus it is consistent with ZFC that the property Split(Λ, Λ) is preserved by unions of less than b subsets of the real line. Introduction In this paper we shall present two directions of applications of semifilters in selection principles on topological spaces. First, we shall consider preservation by unions of the Menger property. Trying to describe the σ-compactness in terms of open covers, K.Menger intro- duced in [Me] the following property, called the Menger property: a topological space X is said to have this property if for every sequence (un)n∈ω of open covers of X there exists a sequence (vn)n∈ω such that each vn is a finite subfamily of un and the collection {∪vn : n ∈ ω} is a cover of X. The class of Menger topological spaces, i.e. spaces having the Menger property appeared to be much wider than the class of σ-compact spaces (see [BT], [CP], [JMSS] and many others), but it has interesting properties itself and poses a number of open questions. One of them, namely the question about the value of additivity of corresponding σ-ideal, arXiv:math/0412498v1 [math.GN] 27 Dec 2004 will be discussed in this paper.
    [Show full text]
  • Arxiv:1603.03361V3 [Math.GN] 18 May 2016 Eeecs R O Eddfrtermidro Hspaper
    PRODUCTS OF MENGER SPACES: A COMBINATORIAL APPROACH PIOTR SZEWCZAK AND BOAZ TSABAN Abstract. We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish pro- ductive properties for versions of Menger’s property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger’s property is strictly between Menger’s and Hurewicz’s classic properties. We include a number of open problems emerging from this study. 1. Introduction A topological space X is Menger if for each sequence U1, U2,... of open covers of the space X, there are finite subsets F1 ⊆ U1, F2 ⊆ U2, . whose union forms a cover of the space X. This property was introduced by Karl Menger [17], and reformulated as presented here by Witold Hurewicz [11]. Menger’s property is strictly between σ-compact and Lindelöf. Now a central notion in topology, it has applications in a number of branches of topology and set theory. The undefined notions in the following example, which are available in the indicated references, are not needed for the remainder of this paper. Example 1.1. Menger spaces form the most general class for which a positive solution of arXiv:1603.03361v3 [math.GN] 18 May 2016 the D-space problem is known [2, Corolarry 2.7], and the most general class for which a general form of Hindman’s Finite Sums Theorem holds [25].
    [Show full text]
  • Perfect Mappings and Their Applications in the Theory of Compact Convex Sets Contents
    Perfect mappings and their applications in the theory of compact convex sets Ji°í Spurný Charles University in Prague, Faculty of Mathematics and Physics Sokolovská 83, 186 75 Prague, Czech Republic Contents 1. Introduction 6 2. Summary of Chapter 2 9 2.1. Perfect images of absolute Souslin and absolute Borel Tychono spaces . 9 2.2. Extending Baire one functions on topological spaces . 17 2.3. Borel sets and functions in topological spaces . 23 3. Summary of Chapter 3 9 3.1. A solution of the abstract Dirichlet problem for Baire one functions . .9 3.2. Descriptive properties of elements of biduals of Banach spaces . 17 4. List of articles (sections) 18 Bibliography 20 Summary We study an interplay between descriptive set theory and theory of compact convex sets. Theory of compact convex sets serves as a general framework for an investigation of Banach spaces as well as more general objects as subsets of Banach spaces, sets of measures etc. Descriptive set theory provides a way how to measure a complexity of given ob- jects. Connections between these two dierent mathematical disciplines provide a useful insight in both of them. We recall that theory of Banach spaces is in a way subsumed by theory of compact convex sets via the following procedure. If E is a Banach space, its dual unit ball BE∗ endowed with the weak* topology is a compact convex set and E can be viewed as an isometric subspace of the space of all ane continuous functions on BE∗ . Further, BE∗ is a natural example of a compact topological space.
    [Show full text]
  • 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]
  • Sailing Over Three Problems of Koszmider
    SAILING OVER THREE PROBLEMS OF KOSZMIDER FELIX´ CABELLO SANCHEZ,´ JESUS´ M.F. CASTILLO, WITOLD MARCISZEWSKI, GRZEGORZ PLEBANEK, AND ALBERTO SALGUERO-ALARCON´ Abstract. We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact KA generated by an almost disjoint family A of infinite subsets of ω — we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin’s axiom the space C(KA) is uniquely determined up to isomorphism by the cardinality of A whenever |A| < c, c while there are 2 nonisomorphic spaces C(KA) with |A| = c. We also investigate Koszmider’s problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of c0 and some C(K) spaces. 1. Introduction Koszmider poses in [26] five problems about the structure of the spaces of continuous functions on the Stone compact KA generated by an almost disjoint family A of infinite subsets of ω. Problem 2, that Koszmider himself solves, is the existence of an almost disjoint family A such that under either the Continuum Hypothesis CH or Martin’s Axiom MA, C(KA) ≃ c0 ⊕ C(KA) is the only possible decomposition of C(KA) in two infinite dimensional subspaces. Very recently Koszmider and Laustsen have obtained in [27] the same result without any additional set-theoretic assumptions. Problem 1 asks whether a similar separable space exists. Argyros and Raikoftsalis [1] call a Banach space X quasi-prime if there exists an infinite dimensional subspace Y such that X ≃Y⊕X is the only possible nontrivial decomposition of X .
    [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]
  • 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]
  • Selective Covering Properties of Product Spaces, II: Gamma Spaces
    SELECTIVE COVERING PROPERTIES OF PRODUCT SPACES, II: γ SPACES ARNOLD W. MILLER, BOAZ TSABAN, AND LYUBOMYR ZDOMSKYY Abstract. We study productive properties of γ spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: (1) Solving a problem of F. Jordan, we show that for every unbounded tower set X ⊆ R of cardinality ℵ1, the space Cp(X) is productively Fr´echet–Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova– Scheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers–Tall, we show that the properties γ and Gerlits–Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel’ski˘ı duality to obtain results concerning local properties of function spaces and countable topo- logical groups. 1. Introduction For a Tychonoff space X, let Cp(X) be the space of continuous real-valued functions on X, endowed with the topology of pointwise convergence, that is, the topology inherited from the Tychonoff product RX . In their seminal paper [14], Gerlits and Nagy characterized the property that the space Cp(X) is Fr´echet–Urysohn—that every point in the closure of a set arXiv:1310.8622v4 [math.LO] 10 Oct 2018 is the limit of a sequence of elements from that set—in terms of a covering property of the domain space X.
    [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]
  • Arxiv:1605.05271V2 [Math.GN] 31 Dec 2016 Property Offtplgclspaces Topological Dorff Obvio Converse the LOTS
    COUNTABLE SUCCESSOR ORDINALS AS GENERALIZED ORDERED TOPOLOGICAL SPACES ROBERT BONNET 1 AND ARKADY LEIDERMAN Abstract. We prove the following Main Theorem: Assume that any contin- uous image of a Hausdorff topological space X is a generalized ordered space. Then X is homeomorphic to a countable successor ordinal (with the order topology). The converse trivially holds. 1. Introduction and Main Theorem All topological spaces are assumed to be Hausdorff. Remind that L is a Linearly Ordered Topological Space (LOTS) whenever there is a linear ordering ≤L on the set L such that a basis of the topology λL on L consists of all open convex subsets. A convex set C in a linear ordering M is a subset of M with the property: for every x<z<y in M, if x, y ∈ C then z ∈ C. The above topology, denoted by λL is called an order topology. Since the order ≤L defines the topology λL on L (but not vice-versa), we denote also by hL, ≤L i the structure including the topology λL. A topological space hX, τ X i is called a Generalized Ordered Space (GO-space) whenever hX, τ X i is homeomorphic to a subspace of a LOTS hL, λL i, that is τ X = λL↾X := {U ∩ X : U ∈ λL} (see [2]). Evidently, every LOTS, and thus any GO-space, is a Hausdorff topological space, but not necessarily separable or Lindel¨of. The Sorgenfrey line Z is an example of a GO-space, which is not a LOTS, and such that every subspace of Z is separable and Lindel¨of (see [4]).
    [Show full text]