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Vladimir V. Tkachuk Special Features of Function Spaces

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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
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