Generalized Metric Spaces and Mappings

Shou LinZiqiu Yun

Atlantis Studies in MathematicsSeries Editor: Jan van Mill

Atlantis Studies in Mathematics

Volume 6

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Jan van Mill, VU University, Amsterdam, The Netherlands

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Shou Lin • Ziqiu Yun

Generalized Metric Spacesand Mappings

Shou LinSchool of Mathematics and StatisticsMinnan Normal UniversityZhangzhouChina

Ziqiu YunDepartment of MathematicsSoochow UniversitySuzhouChina

ISSN 1875-7634 ISSN 2215-1885 (electronic)Atlantis Studies in MathematicsISBN 978-94-6239-215-1 ISBN 978-94-6239-216-8 (eBook)DOI 10.2991/978-94-6239-216-8

Library of Congress Control Number: 2016951688

Translation and revision from the Chinese language edition: 广义度量空间与映射 by Shou Lin© Science Press, Beijing 2007. All Rights Reserved.© Atlantis Press and the author(s) 2016This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by anymeans, electronic or mechanical, including photocopying, recording or any information storage andretrieval system known or to be invented, without prior permission from the Publisher.

Printed on acid-free paper

Foreword

What are generalized metric spaces? Most often, this expression denotes a systemof classes of topological spaces, each of which is defined with the help of sometypical topological property of metrizable spaces. For example, the classes ofparacompact spaces, Moore spaces, spaces with a point-countable base, sub-metrizable spaces, perfectly normal spaces, first-countable spaces belong to thissystem. We also include in it the class of symmetrizable spaces and the class of D-metrizable spaces. The topologies of these spaces are generated by generalizedmetrics. In this way, we obtain a rich system of classes of topological spaces whichare all emerging, growing and spreading in many directions, from the same pow-erful germ—the concept of the topology generated by a metric.

The system of generalized metrizable spaces (below, we often use an abbrevi-ation “GMS-system” to denote it) also includes various relations between itsmembers. The simplest among them is, of course, the inclusion relation. But muchdeeper, often unpredictable links between classes of topological spaces are estab-lished by means of mappings. Mappings of various kinds serve as instruments bymeans of which the abstract geometric objects, such as topological spaces, topo-logical groups, metric spaces, function spaces and so on, can be compared betweenone another and further be classified. This can be done at the level of individualspaces, but similarly, natural classes of mappings can be used to establish funda-mental connections between classes of topological spaces.

The ideas and problems of the theory of generalized metrizable spaces, in par-ticular, the general metrization problem, greatly influenced all domains ofset-theoretic topology. The goal of this book is to describe the present structureof the theory of generalized metric spaces and the modern tendencies in this theory.An important fact: this is the first book on generalized metrizable spaces. Somebrilliant survey papers on this topic were written by prominent topologists R.Hodel, G. Gruenhage, D. Burke, D. Lutzer, H.H. Wicke and J. Worrell, W.

v

Fleissner, G.M. Reed who themselves contributed greatly to the field. But a book onthis rapidly developing subject was lacking. It should have appeared at least 20years earlier. This has happened in China, where a version of this book hasappeared in Chinese in 1995. So, it is not astonishing that a strong input in theset-theoretic topology was made by in the last 20 or 30 years by Guoshi Gao,Zhimin Gao, Shou Lin, Chuan Liu, Fucai Lin, Ziqiu Yun, Liang-Xue Peng, LeiMou, Wei-Xue Shi, Li-Hong Xie, Jing Zhang and many others.

It is an excellent news that, finally, an expanded and polished internationaledition of this book will appear in English. The contents are well selected. The bookhas three chapters and two appendices. The main part, consisting of the threechapters, is written in a very systematic way. Theorems and examples form a partof the big picture; they are supplied with detailed proofs. Stratifiable spaces, spaceswith a r-discrete network, bases of countable order and their theory developed byH.H. Wicke and J. Worrell, uniform bases introduced by P.S. Alexandroff,monotonically normal spaces of R.W. Heath, p-spaces and their theory, quotients ofseparable metrizable spaces, A.H. Stone’s theorem on paracompactness of metricspaces, are presented with a great force. All major theorems obtained in the past 50years in GMS-theory, in particular, V.I. Ponomarev’s theorem characterizingfirst-countable spaces as open continuous images of metric spaces, V.V. Filippov’stheorem on metrizability of paracompact p-spaces with a point-countable base,generalizing A.S. Miščenko’s theorem on compacta with such a base, Michael’stheorem on preservation of paracompactness by closed continuous mappings, acharacterization of perfect preimages of metrizable spaces as paracompact p-spaces(by A.V. Arhangel’skiǐ), another, independent, characterization of the same spacesby K. Morita (as paracompact M-spaces), all these basic results of the GMS-theorythe reader will find in the book. Appendices A and B are written in the form ofsurvey, they provide the reader with many further advanced and more special factsand open questions, with metamathematical discussions and attractive philosophicalinsights, omitting detailed proofs. Much of additional information on the historyof the subject is also provided here. An attractive feature of this book is its inter-national character. The list of references is very rich, it includes about 500 entries,and will be of great value for readers. Besides the papers of mathematicians wehave already mentioned above, the list includes the classical ground-breakingpapers on GMS-theory, written by R. Bing, C. Dowker, J. Nagata, E. Michael, M.E.Rudin, A.H. Stone, Yu.M. Smirnov, K. Nagami, Z. Frolík, R. Engelking, Z.Balogh, J. van Mill, M. Hušek, H. Tamano, P. Nyikos, J. Chaber, S. Nedev, C.Borges, T. Hoshina, R. Stephenson, Y. Tanaka, H. Junnila, N.V. Veličko, M. Itō,D. Shakhmatov, M. Sakai, Y. Yajima and others.

The book, written by Professors Shou Lin and Ziqiu Yun who are active expertsin the GMS-theory with a long list of original contributions, will be a most valuablesource of information for graduate and undergraduate topology students wishing to

vi Foreword

start their own research in this field or to apply it to other mathematical disciplines.It will be also very helpful to experts in topology already working in GMS-theory,as well as to those who want to enrich their research in other topics by linking it toGMS-theory. The book will also help to develop new original special courses ingeneral topology.

April 2016 Alexander V. Arhangel’skiǐDistinguished Professor Emeritus

Ohio University, and Professor of Moscow University

Foreword vii

Preface

The uniqueness of this book lies in describing generalized metric spaces by meansof mappings. Back in 1961, Alexandroff [3] proposed the idea of using the mappingmethod to study spaces in the first Prague Topological Symposium. The surveypaper “Mappings and spaces” written by Arhangel’skiǐ [31] in 1966 inherited anddeveloped the idea. We were greatly interested in this paper. Professors L. Wu andB. Chen then translated it into Chinese (originally in Russian) and published it in“Mathematics” (1981–1982), and wanted to arouse the interest of Chinese scholars.For a comprehensive introduction to the theory of generalized metric spaces, werecommend the books written by Burke, Lutzer [88] and Gruenhage [162], and twochapters written by Nagata [378] and Tamano [449] in the book “Topics in GeneralTopology” [364].

There are roughly three perspectives of investigating spaces by using mappings:

(1) Which classes of generalized metric spaces can be represented as images orpreimages of metric spaces under certain mappings? For example, theM-spaces introduced by Morita [360] for investigating the normality of pro-duct spaces can be expressed as preimages of metric spaces underquasi-perfect mappings. This has opened up a new way of investigatingM-spaces and established connection between this class of spaces and metricspaces.

(2) What are the intrinsic characterizations of images of metric spaces undercertain mappings? For example, the closed images of metric spaces (usuallycalled Lašnev spaces) are characterized, by Foged [130], as regularFréchet-Urysohn spaces with a r-hereditarily closure-preserving k-network.Thus, it can be compared with the Burke-Engelking-Lutzer metrizationtheorem [87] and one can connected these spaces with some generalizedmetric spaces defined by k-networks, for example @-spaces, etc.

(3) Certain generalized metric spaces are preserved under what kinds of map-pings? Take the class of metrizable spaces as an example, by theHanai-Morita-Stone theorem [363, 441], we know that metrizability isinvariant under perfect mappings. Michael [336] further proved that

ix

metrizability is invariant under countably bi-quotient closed mappings, whichshows the charm of bi-quotient mappings.

These simple examples in the above three aspects reveal great colorfulness andattractiveness of infiltrations of mapping method for investigating generalizedmetric spaces.

Following the ideas and methods of Alexandroff-Arhangel’skiǐ, this book pre-sents the relevant results in the theory of generalized metric spaces in the past threedecades from the perspective of mappings particularly the achievements of Chinesescholars in recent years. As an interesting scientific research reading material, itfurther elaborates these ideas and methods.

This book is also suitable as an elective course material or a teaching referencebook for the graduates and senior undergraduates of mathematics major, or as areading material for graduate students of general topology. It also can be used asa reference book for mathematicians and scientific researchers in other fields.

Suzhou, China Guoshi GaoSeptember 1992

x Preface

Preface to the English Edition

The concept of paracompactness introduced by Dieudonné [112] in 1944 is asignificant sign of general topology to enter the peak period. The extraordinaryBing-Nagata-Smirnov metrization theorem [62, 370, 428], established in the years1950 and 1951, created a fundamental change for the full exploration of the natureof metrizability. Meanwhile, the theorem disclosed a bright future of the investi-gation on properties of generalized metric spaces. The deep study on paracom-pactness and metrizability performs the prelude to the research on the theory ofgeneralized metric spaces.

In 1961, at the international topological symposium named “General Topologyand its Relationship with Modern Analysis and Algebra” in Prague, Alexandroff [3]put forward the idea of investigating spaces by mappings, namely, to connectvarious classes of spaces by using mappings as a linkage. In this way, the funda-mental research framework and whole structure of the theory of topological spacescan be reflected by the relationships between mappings and spaces. Mappings thusbecame a powerful tool for revealing the internal properties of various classes ofspaces. In 1966, Arhangel’skiǐ [31] published the historical literature titled“Mappings and Spaces” which presented a series of constructive concrete steps ofhow to operate the Alexandroff idea. As a milepost in the road of the vigorousdevelopment of general topology, it ushered in an innovation era for investigatingspaces by mappings. Since then, the Alexandroff idea became an indispensablemethod for the research. It promoted a rapid development of general topology,particularly the theory of generalized metric spaces. In a word, the Alexandroff ideaof mutual classifications of spaces and mappings has constituted an important partof contemporary general topology [4].

According to Alexandroff and Arhangel’skiǐ, the core of investigating spaces bymeans of mappings is to establish the extensive connections, with the aid ofmappings, between the class of metric spaces and classes of spaces having specifictopological properties, to study the intrinsic characterizations of images of metricspaces under various mappings, and to discuss which kinds of mappings can pre-serve certain classes of spaces. This framework determines the goals of this book: to

xi

comprehensively describe characterizations of images of metric spaces under var-ious mappings and to establish mapping theorems for several important generalizedmetric spaces.

The most prominent feature of this book is to discuss the theory of generalizedmetric spaces in a systematic way from the perspective of mappings. Through themapping theorems of generalized metric spaces, this book tries to point out that theprinciple of classifying spaces by mappings is undeniably a powerful research toolthat has decisive significance in general topology, by which one can peep the fulllinkage between spaces and mappings. Most of the contents of this book are fromhundreds of research papers on spaces and mappings published in the recent50 years. While paying attention to the innovation and unique features of thecontents, this book highlights the outstanding achievements of Chinese scholars inrecent years. The authors hope this book can guide readers to grasp the principleof classifying spaces by mappings and catch up the new developments of the theoryof generalized metric spaces. We also hope this book can provides solid support forreaders in their further research work and can magnify the impacts from China’sgeneral topological circles to the world.

This book is composed of three chapters, two appendices and a list of more than400 references. To provide necessary preparation for the descriptions of relation-ships between spaces and mappings in the following two chapters, Chap. 1 brieflyintroduces some basic concepts of generalized metric spaces, the properties offundamental operations on these spaces and their simple characterizations. Bymeans of several families of sets with specific properties and the concepts of basesor their generalizations, Chap. 2 presents the intrinsic characterizations of images orpreimages of metric spaces under quotient mappings, pseudo-open mappings,countably bi-quotient mappings, open mappings, closed mappings andcompact-covering mappings or under these mappings with some additional con-ditions, for example the fibers are separable or compact. Chapter 3 introducesseveral classes of generalized metric spaces defined by specific bases, weak bases,k-networks, networks and (mod k)-networks etc., such as M-spaces, p-spaces,g-metrizable spaces, @-spaces, k-semi-stratifiable spaces, r-spaces, R-spaces and soon. Characterizations and mapping theorems of these spaces are also obtained inthis chapter. There are two appendices at the end of this book. The first one aims tohelp readers to better understand some results on the covering properties used in thetext. The second one aims to help readers to obtain a more comprehensive visionof the theory of generalized metric spaces, particularly to help them clearly rec-ognize the place of Alexandroff’s idea in contemporary general topology.

The first two editions of this book were written in Chinese and published in 1995and 2007, respectively [271, 282]. They have played an active role on the devel-opment of the theory of spaces and mappings. The primary goal of this new editionis to update recent developments in the area and to continue to revise the text for thesake of clarity and accessibility for readers. In order to arouse interest about thetheory of generalized metric spaces and promote contributions of Chinese mathe-maticians in a broader international society, this edition is published in English.

xii Preface to the English Edition

We benefited from the input of many talented people in writing this book.The most important person in our academic career is Professor G. Gao.1 He is ourmentor. His outstanding work in spaces and mappings and his actively promotingAlexandroff-Arhangel’skiǐ ideas have created opportunities for us to deeply engagein the study of this subject [140, 142]. We are very grateful to Prof. Gao’s thoroughcoaching, invaluable advice, altruistic helps and constant encouragements since1979. He has laid down the learning and research foundations for us, pointed outthe direction of further exploration to us and given us courage and confidence in ourresearch. This book reflects Professor Gao’s research style and academic thinking toa certain extent. We also thank Professor Alexander V. Arhangel’skiǐ for hisvaluable comments on improving the writing of this book.

This edition is supported by the fund projects “The three spaces problems inparatopological groups” and “The coverage problems in sensor networks based onlocal information and rough set theory” of the National Natural Science Foundationof China (Project Numbers 11471153 and 61472469).

Zhangzhou, China Shou LinSuzhou, China Ziqiu YunMay 2016

1G. Gao, see “Biographies of Modern Chinese Mathematicians in the 20th Century, V. 3”, editedby M. Chen, Jiangsu Education Press, Nanjing, 1998, 287–297.

Preface to the English Edition xiii

Contents

1 The Origin of Generalized Metric Spaces . . . . . . . . . . . . . . . . . . . . . . 11.1 Notations and Terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Stratifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Networks and (mod k)-Networks . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 k-Networks and Weak Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7 Generalized Countably Compact Spaces . . . . . . . . . . . . . . . . . . . . 341.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Mappings on Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.1 Classes of Mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.2 Perfect Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.3 Quotient Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.4 Open Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.5 Closed Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.6 Compact-Covering Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.7 s-Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.8 ss-Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.9 p-Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232.10 Compact Mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1302.11 r-Locally Finite Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3 Generalized Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.1 Spaces with Point-Countable Covers . . . . . . . . . . . . . . . . . . . . . . 1483.2 R-Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.3 r-Spaces and Semi-Stratifiable Spaces . . . . . . . . . . . . . . . . . . . . . 1803.4 k-Semi-Stratifiable Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923.5 Mi-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2033.6 Developable Spaces and p-Spaces . . . . . . . . . . . . . . . . . . . . . . . . 212

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3.7 M-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2253.8 @-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2343.9 g-Metrizable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2473.10 Several Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Appendix A: Characterizations of Several Covering Properties . . . . . . . 259

Appendix B: The Formation of the Theory of GeneralizedMetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

xvi Contents

Chapter 1The Origin of Generalized Metric Spaces

Metrization theory is the core in the study of general topology and the theory ofgeneralized metric spaces is a generalization of this theory. By the opinion of Burkeand Lutzer [88], the theory of generalized metric spaces has its roots mainly inthree problems: the metrization problem, the problem of paracompactness in productspaces and the mutual classification problem of spaces and mappings suggested byAlexandroff in 1961 [3].

What is a “generalized metric space”? Perhaps, all the topological propertieswhich are weaker than those for metrization can be called generalized metric prop-erties. However, this interpretation is too broad. Roughly speaking, the class of gen-eralized metric spaces are such classes of spaces that they posses some propertieswhich are useful in characterizing metrizability, and inherit many delicate natures ofmetric spaces, and some of the theory or techniques of metric spaces can be carriedover to these classes [162, 193].

Hodel [193] pointed out: “There are a number of reasons why generalized metriz-able spaces are worthy of study. Perhaps the most important reason is that suchclasses increase our understanding of the metrizable spaces. But in addition, topolo-gists are continually seeking broader classes of spaces for which especially importantresults hold.” For example, the Katětov–Morita dimension theorem, the Dugundjiextension theorem and the Borsuk homotopy extension theorem etc. can be estab-lished in these spaces. Because of these reasons, the theory of generalized metricspaces has been one active research direction in general topology since the 1960s. Alarge number of problems of this theory which are formed by the mutual blendingof other branches, such as axiomatic set theory, mathematical logic, combinatoricsmathematics, functional analysis, topological algebra, dynamic system and computerscience etc., have been included in the books “Open Problems in Topology” [342],“Open Problems in Topology II” [392] and “Problems from Topology Proceedings”[391]. The achievements of the theory of generalized metric spaces between 1960and 2015 have been summarized in some important works, such as Arhangel’skiı̌[31, 34], Burke and Lutzer [88], G. Gao [140, 141], Gruenhage [162, 163, 164, 166],

© Atlantis Press and the author(s) 2016S. Lin and Z. Yun, Generalized Metric Spaces and Mappings,Atlantis Studies in Mathematics 6, DOI 10.2991/978-94-6239-216-8_1

1

2 1 The Origin of Generalized Metric Spaces

Hodel [193], Kodama andNagami [230], S. Lin [282, 285],Morita andNagata [364],Nagata [373, 376]. Numbers of challenging problems proposed by many scholars,together with some of long-term unresolved problems, have become the birthplacesof further development of the theory of generalized metric spaces. Just as Hodel[193] pointed out after summing up the staggering achievement of the theory ofgeneralized metric spaces: “But more important perhaps is the fact that the study ofgeneralized metrizable spaces is by no means complete; rather, it continues to growwith many new and important results appearing every year.”

In this chapter, we derive most of the generalized metric spaces that will bediscussed in this book from three perspectives: distance functions, bases or theirextensions and generalized countable compactness. At the same time, we give somesimple characterizations of these spaces, describe their basic operation properties,such as hereditary properties, productive properties and properties preserved undertopological sums, and discuss their preliminary relationships with some other classesof spaces.

1.1 Notations and Terminologies

Through the Book, Spaces are All Hausdorff Topological SpacesIn this section, we first give definitions of the notations and terminologies usedfrequently in this book, and then list several classic results.

1.1.1 Sets of Real Numbers and Cardinal Numbers

We denote the real line by R. The sets of natural numbers, positive integers, ratio-nal numbers, irrational numbers and nonnegative real numbers are denoted as ω,N, Q, P and R+ respectively. The letter ω also denotes the smallest infinite ordinal.The unit closed interval is denoted by I. Denote S1 = {0} ∪ {1/n : n ∈ N}. The car-dinalities ofN andR are denoted by ℵ0 and c respectively. The smallest uncountablecardinality (resp. ordinal) is denoted as ℵ1 (resp. ω1).

1.1.2 Operations on Subsets of Topological Spaces

The topology and the set of closed subsets of a space X are denoted by τ(X) andτ c(X) respectively, and when there is no ambiguity, τ(X) and τ c(X) are denoted asτ and τ c respectively. Let A and (Y, τ ′) be a subset and a subspace of X respectivelyand let Z ⊂ Y . Then

1.1 Notations and Terminologies 3

the closure of A in X is denoted as A or cl(A);the interior of A in X is denoted as A◦ or int(A);the boundary of A in X is denoted as ∂ A;the set of accumulation points of A in X is denoted as Ad ;the closure of Z in Y is denoted as clY (Z) or clτ ′(Z);the interior of Z in Y is denoted as intY (Z) or intτ ′(Z).

1.1.3 Families of Sets in Topological Spaces

In the sequel, a family always means a family of sets. For any space X , denote

K (X) = {K ⊂ X : K is a compact set in X}, andS (X) = {S ⊂ X : S is a convergent sequence in X with its limit point}.

Each finite subset of X is regarded as a trivial convergent sequence. A sequence {xn}in X is called a nontrivial sequence, if xn �= xm whenever n �= m.

Let P be a family in X . Then

P

4 1 The Origin of Generalized Metric Spaces

1.1.4 Mappings on Spaces

Assume that X and Y are spaces and f : X → Y is a mapping.Suppose A ⊂ X , the restriction of f to A to be the function f|A : A → f (A)

whose rule is f|A(x) = f (x), for each x ∈ A.For each B ⊂ Y , f| f −1(B) is denoted by fB and called the restriction of f to B.If P is a family in X , then f (P) = { f (P) : P ∈ P} is called the image of P

under f .If F is a family in Y , then f −1(F ) = { f −1(F) : F ∈ F } is called the inverse

image of F under f or the preimage of F under f .Suppose X, Y, Z and W are spaces. If f : X → Y, g : X → Z , and h : W → Z ,

then the diagonal mapping f Δg : X → Y × Z and the product mapping

f × h : X × W → Y × Z

are defined as ( f Δg)(x) = ( f (x), g(x)) and ( f × h)(x, w) = ( f (x), h(w)) respec-tively. The mappings Δα∈Γ fα : X → ∏α∈Γ Yα and

∏α∈Γ fα :

∏α∈Γ Xα →∏

α∈Γ Yα can be defined similarly.The identity mapping from X onto X is denoted by idX .For the product space

∏α∈Γ Xα and any β ∈ Γ , the projection of

∏α∈Γ Xα to

the β-th coordinate space Xβ is denoted as πβ : ∏α∈Γ Xα → Xβ .

1.1.5 Operations of Spaces

Suppose Φ is a topological property.

(1) The property Φ is said to be additive if the topological sum⊕

α∈Λ Xα has Φwhenever {Xα}α∈Λ is a family of spaces having Φ.

(2) The propertyΦ is said to be hereditary (resp. open hereditary, closed hereditary)if any subspace (resp. open subspace, closed subspace) of X has Φ wheneverX has Φ.

(3) The property Φ is said to be productive (resp. finitely productive, countablyproductive) if the product space

∏α∈Λ Xα has Φ whenever {Xα}α∈Λ is a family

(resp. finite family, countable family) of spaces with the property Φ.(4) The property Φ is said to be preserved or invariant (resp. inversely preserved or

inversely invariant) under a class of mappingsL , provided that if f : X → Y isan onto mapping with f ∈ L and X (resp. Y ) has the property Φ, then Y (resp.X ) also has Φ.

For the sake of convenience, we will use terms “Φ space”, “Φ property” and “Φproperty of space” without distinction in the sequel.

1.1 Notations and Terminologies 5

1.1.6 Several Classic Results

For future reference, we list in this section several classic lemmas and theorems ingeneral topology [119] which will be used in the subsequent sections.

(1) The Zermelo theorem on well-ordering. On every set X there exists a relationwhich well-orders X .

(2) The continuum hypothesis (abbreviated as CH). c = ℵ1.(3) The Urysohn-Tychonoff metrization theorem. Every regular space with a count-

able base can be embedded in the Hilbert cube Iℵ0 , and hence is a metrizablespace.

(4) The Urysohn lemma. A space X is a normal space if and only if for everypair A, B of disjoint closed subsets of X , there exists a continuous functionf : X → I such that f (A) ⊂ {0} and f (B) ⊂ {1}.

(5) The Urysohn extension theorem. A space X is a normal space if and only if everyreal valued continuous function from a closed subspace of X is continuouslyexpandable over X .

(6) The Tychonoff theorem. Every Cartesian product of compact spaces is compact.(7) The Tychonoff compact extension theorem. A space is a completely regular

space if and only if there exists a compact extension of it.(8) The Baire category theorem. The spaceR is second category, i.e. the intersection

of countably many dense open subsets of R is dense in R, and hence R is notthe union of countably many nowhere dense closed subspaces of R.

(9) The diagonal theorem. If every fα : X → Yα is continuous mapping andthe family { fα}α∈Λ separates points from closed sets in X , then the diagonalΔα∈Λ fα : X → ∏α∈Λ Yα is an embedding, i.e. Δα∈Λ fα : X → Δα∈Λ fα(X) isa homeomorphic mapping.

1.2 Distance Functions

One of the reasons for metric spaces being accepted easily by mathematicians is theexistence of a metric. By distance functions, we introduce in this section the conceptsof metric spaces, symmetric spaces, semi-metric spaces and relevant developablespaces, quasi-developable spaces. The Stone theorem is proved in this section.

To prove proposition “(A) ⇒ (B) for a space X”, sometimes we assume thatthe condition of (A) is true in X . In the sequel, we shall do in that way without anyexplanation.

Definition 1.2.1 A function d : X × X → R+ is called a distance (resp. pseudo-distance) on X if d satisfies the following conditions for all x, y, z ∈ X :(1) d(x, y) = 0 if and only if x = y (resp. d(x, x) = 0),(2) d(x, y) = d(y, x),(3) d(x, y) � d(x, z) + d(z, y).

6 1 The Origin of Generalized Metric Spaces

For every A, B ⊂ X and x ∈ X , define

d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B};d(x, B) = d(B, x) = d({x}, B).

For each ε > 0, letB(x, ε) = {y ∈ X : d(x, y) < ε}.

Then X is called a metric space (resp. pseudo-metric space) if X is a topologicalspace generated by the base {B(x, ε) : x ∈ X, ε > 0}. This topology is also calledthe metric topology generated by d. If the topology for X is generated by a distanced, then X is called a metrizable space and d is called a metric on X .

Suppose {(Xα, dα)}α∈Λ is a family of metric spaces. Let X = ⊕α∈Λ Xα anddefine d : X × X → R+ as follows:

d(x, y) ={min{dα(x, y), 1}, there exists α ∈ Λ such that x, y ∈ Xα,1, otherwise.

Then (X, d) is a metric space and d is called the standard topological sum metric.Metric spaces are general mathematics research objects which have many good

properties. For example, metric spaces are additive, hereditary and countably pro-ductive. In the theory of metric spaces, the A.H. Stone theorem is one of the mostimportant results.

Definition 1.2.2 ([62]) A family P in a space X is said to be discrete at a pointx ∈ X if there is a neighborhood V of x such that V meets at most one element ofP . If P is discrete at every point x ∈ X , then P is said to be discrete in X . Theunion of countably many discrete families is called a σ -discrete family. Generally,if Φ is a property of families, then the union of countably many families having Φis called a σ -Φ family.

Theorem 1.2.3 [440] (The Stone theorem) Every metric space is a paracompactspace.

Proof Suppose {Uα}α∈Λ is an open cover of a metric space (X, d). For every α ∈ Λand n ∈ N, define(3.1) Uα,n = {x ∈ X : B(x, 1/2n) ⊂ Uα}.

Then Uα = ⋃n∈N Uα,n , and x ∈ Uα,n if and only if d(x, X − Uα) � 1/2n .Hence,

(3.2) if x ∈ Uα,n and y /∈ Uα,n+1, then d(x, y) > 1/2n+1.By theZermelo theoremonwell-ordering,wemay assume< is awell-orderingon Λ. Let

(3.3) U ∗α,n = Uα,n −⋃

γ

1.2 Distance Functions 7

(3.4) U ∗β,n ⊂ X − Uα,n+1 or U ∗α,n ⊂ X − Uβ,n+1.If x ∈ U ∗α,n and y ∈ U ∗β,n , then by (3.3) and (3.4), when α < β, x ∈ Uα,n andy /∈ Uα,n+1; when β < α, y ∈ Uβ,n and x /∈ Uβ,n+1. Thus, we always haved(x, y) > 1/2n+1 by (3.2), i.e.

(3.5) d(U ∗α,n, U ∗β,n) � 1/2n+1.For each x ∈ X , let α be the least element in Λ such that x ∈ Uα . Take n ∈ Nsuch that x ∈ Uα,n . By (3.3), x ∈ U ∗α,n , and it means that

(3.6)⋃

α∈Λ,n∈N U ∗α,n = X .For every α ∈ Λ and n ∈ N, define

(3.7) U+α,n = {x ∈ X : d(x, U ∗α,n) < 1/2n+3}.Then

(3.8) U ∗α,n ⊂ U+α,n ⊂ Uα .By (3.5), (3.7) and the triangle inequality, it is easy to prove d(U+α,n, U

+β,n) �

1/2n+2 for every α, β ∈ Λ with α �= β. So for each x ∈ X , B(x, 1/2n+3) meets atmost one element of {U+α,n}α∈Λ, and it follows that {U+α,n}α∈Λ is a discrete family inX consisting of open sets. Hence, {U+α,n}α∈Λ,n∈N is a σ -discrete open refinement of{Uα}α∈Λ, and that proves X is a paracompact space. �Definition 1.2.4 A function d : X × X → R+ is called a symmetric [382] on a setX if for every x, y ∈ X ,(1) d(x, y) = 0 if and only if x = y;(2) d(x, y) = d(y, x).A space (X, τ ) is said to be a symmetrizable space [7] if there is a symmetric d on Xsuch that the topology τ given on X is generated by the symmetric d, that is, a subsetU ∈ τ if and only if for each x ∈ U , there exists ε > 0 such that B(x, ε) ⊂ U .

Suppose d is a symmetric on X . Then a necessary and sufficient condition for(X, d) being a symmetrizable space is that for any A ⊂ X , A is closed in X if andonly if for each x ∈ X − A, d(x, A) > 0. It is easy to verify that symmetrizabilityis an additive, open hereditary and closed hereditary property.

A space X is called anℵ1-compact space [214] if every discrete closed subspace ofX is countable. Obviously, Lindelöf spaces and closed hereditarily separable spacesare ℵ1-compact spaces, and every locally finite family in an ℵ1-compact space iscountable.

Theorem 1.2.5 ([380, 381]) Every symmetrizable ℵ1-compact space is a hereditar-ily Lindelöf space.

Proof Suppose (X, d) is a symmetrizable ℵ1-compact space. To prove X is a hered-itarily Lindelöf space, we only need to prove every open subspace of X is a Lindelöfspace. Suppose Y is an open subset of X an U = {Uα : α ∈ Λ} is an open cover ofY , where Λ is a well-ordering set. For every n ∈ N and α ∈ Λ, let

Mnα = {x ∈ Y : x ∈ Uα −⋃

β

8 1 The Origin of Generalized Metric Spaces

If Mnα �= ∅, then pick yα,n ∈ Mnα . Let Dn = {yα,n : α ∈ Λ}.Claim. Dn is a closed discrete set in the subspace Y .Let E ⊂ Dn and x ∈ Y − E . Take λ ∈ Λ and m � n such that x ∈ Uλ −⋃β

1.2 Distance Functions 9

Theorem 1.2.8 The following are equivalent:

(1) X is a semi-metrizable space.(2) There is a semi-metrizable function on X, i.e. there is a g-funct-

ion on X satisfying the following condition: for any x ∈ X and any sequence{xn} in X with xn ∈ g(n, x) or x ∈ g(n, xn), xn → x [176].

(3) There is a semi-development for X [1].(4) X is a first countable (or Fréchet–Urysohn) symmetrizable space [31].

Proof (1) ⇒ (3). Suppose X is a semi-metrizable space. For each n ∈ N, let

Fn = {A ⊂ X : diamA < 1/n},

where diamA = sup{d(x, y) : x, y ∈ A}. Then for each x ∈ X , st(x,Fn) =B(x, 1/n). So {st(x,Fn)}n∈N is a neighborhood base of x .

(3) ⇒ (2). Suppose {Fn} is a semi-development for X . We may assume thatFn+1 refines Fn . Define g : N × X → τ by g(n, x) = st(x,Fn)◦. Then g is asemi-metrizable function on X .

(2) ⇒ (4). Suppose g is a semi-metrizable function on X . Obviously, X is a firstcountable space. For every x, y ∈ X with x �= y, let

m(x, y) = min{n ∈ N : y /∈ g(n, x), x /∈ g(n, y)}.

Define d : X × X → R+ as follows:

d(x, y) ={0, x = y,1/m(x, y), x �= y.

Then d is a semi-metric on X .(4) ⇒ (1). Suppose X is a Fréchet–Urysohn space and d is a symmetric on X . If

A ⊂ X and x ∈ A − A, then there is a sequence {xn} in A converging to x . Assumed(xn, x) � 0. Take a subsequence Z of {xn} such that d(x, Z) > 0. Let T = Z ∪{x}.Then T is a closed subspace of X . It follows that (T, d) is a symmetrizable space,and hence x is an isolated point in T , a contradiction. Thus, d(xn, x) → 0, sod(x, A) = 0. �Corollary 1.2.9 Semi-metrizability is countably productive.

Proof Suppose gn is a semi-metrizable function on Xn for each n ∈ N. Define

g : N ×(

∏

n∈NXn

)→ τ

(∏

n∈NXn

)

by g(m, x) = (∏n�m gn(m, xn)) ×∏

n>m Xn , where x = (xn) ∈∏

n∈N Xn . Then gis a semi-metrizable function on

∏n∈N Xn . �

10 1 The Origin of Generalized Metric Spaces

Definition 1.2.10 ([113]) A neighborhood assignment for a space (X, τ ) is a func-tion φ : X → τ satisfying x ∈ φ(x) for each x ∈ X . X is called a D-space if, foreach neighborhood assignment φ for X , there is a discrete closed set D in X suchthat {φ(d) : d ∈ D} covers X .

van Douwen and Pfeffer [114] raised several interesting questions on D-spaces.Among them, the question that whether every regular Lindelöf space is a D-space isstill open. However, numbers of generalizedmetric spaceswere found to be D-spaces[36, 165].

Theorem 1.2.11 ([84]) Symmetrizable spaces are hereditarily D-spaces.

Proof Suppose (Z , d) is a symmetrizable space and X ⊂ Z with the subspacetopology τ . To show X is a D-space, let φ : X → τ be a neighborhood assignmentfor X and let U = {φ(x) : x ∈ X}. Since open subspaces of symmetrizablespaces are symmetrizable, we may assume each φ(x), x ∈ X , is actually open inZ and Z = ⋃x∈X φ(x). We need to find a closed discrete subset D of X such thatX ⊂ ⋃x∈D φ(x).

For each x ∈ X let kx ∈ N such that B(x, 1/kx ) ⊂ φ(x). For each n ∈ N, letIn = {x ∈ X : kx = n}. Well-order X so that when y ∈ Ii , z ∈ I j for every i < jthen y < z.

Now, recursively find Jn ⊂ In as follows: for each x ∈ In ,

x ∈ Jn ⇔ x = min((In − ∪{φ(y) : i < n, y ∈ Ji }) − ∪{φ(y) : y ∈ Jn, y < x}).

Let D = ⋃n∈N Jn .Claim 1. W = {φ(x) : x ∈ D} covers X .For any y ∈ X find m ∈ N such that y ∈ Im . If y ∈ Jm , then certainly y ∈ φ(y) ⊂

∪W . If y /∈ Jm , then

y ∈ (⋃

{φ(x) : i < m, x ∈ Ji }) ∪ (⋃

{φ(x) : x ∈ Jm}) ⊂ ∪W .

Claim 2. D is a closed discrete set in ∪W (and hence in X ).This follows if we show that for any t ∈ D, D − {t} is closed. To this end

we may assume Z = ∪W and let x ∈ Z − (D − {t}). It suffices to prove thatB(x, 1/n)∩ (D −{t}) = ∅ for some n ∈ N. Let m be the first element of N such thatthere is a first element y of Jm with x ∈ φ(y). Now, for all z ∈ D with y < z we havez /∈ φ(y). Also, for all z ∈ D with z < y we have B(z, 1/m) ⊂ B(z, 1/kz) ⊂ φ(z)and x /∈ φ(z), so z /∈ B(x, 1/m). That is, φ(y) ∩ B(x, 1/m) ∩ (D − {y}) = ∅, thenB(x, 1/n)∩(D−{y}) = ∅ for some n ∈ N. If t = y, thenwe are done. If t �= y, thensince y ∈ D we know x �= y and there is j ∈ N such that y /∈ B(x, 1/j). This givesthat B(x, 1/j)∩φ(y)∩ B(x, 1/m)∩(D−{t}) = ∅, then B(x, 1/n)∩(D−{t}) = ∅for some n ∈ N. �Definition 1.2.12 A sequence {Un} of open covers on X is called a developmentfor X if it is a semi-development for X . A space with a development is called a

1.2 Distance Functions 11

developable space [8] and a regular developable space is called a Moore space [357].A sequence {Un} of families consisting open sets in X is called a quasi-developmentfor X if for every x ∈ U ∈ τ , there is n ∈ N such that x ∈ st(x,Un) ⊂ U . A spacewith a quasi-development is called a quasi-developable space [55].

A space is called a perfect space if each closed set in this space is a Gδ-set.

Theorem 1.2.13 The following are equivalent:

(1) X is a developable space.(2) There is a developable function on X, i.e. there is a g-function on X satisfying

the following condition: for each x ∈ X and any sequences {xn}, {yn} in X with{x, xn} ⊂ g(n, yn), xn → x [176].

(3) X is a quasi-developable perfect space [55].

Proof (1)⇒ (2). Suppose {Un} is a development for X . For every x ∈ X and n ∈ N,take Un ∈ Un such that x ∈ Un . Let g(n, x) = ⋂i�n Ui . Then g is a developablefunction on X .

(2) ⇒ (3). Suppose g is a developable function on X . Obviously, for any closedsubset A of X , A = ⋂n∈N g(n, A), and hence X is perfect. For every n ∈ N, letUn = {g(n, x) : x ∈ X}. Then {Un} is a development, hence a quasi-developmentfor X .

(3) ⇒ (1). Suppose {Un} is a quasi-development for X and X is a perfect space.For each n ∈ N, there is a sequence {Fn, j } j∈N of closed sets in X such that ∪Un =⋃

j∈N Fn, j . Let Hn, j = Un ∪ {X − Fn, j }. Then {Hn, j }n, j∈N is a development forX . �

Developable spaces and quasi-developable spaces are additive, hereditary andcountably productive. Every developable space is a semi-metric subparacompactspace (see Theorem A.3.3 in Appendix A). The following theorem gives a simplesufficient condition for a semi-metric space to be a developable space. A baseB fora space X is said to be point-countable if each point of X only belongs to at mostcountably many elements ofB.

Theorem 1.2.14 ([178]) Every semi-metric space with a point-countable base is adevelopable space.

Proof Suppose U is a point-countable base for a semi-metric space X and � is awell-ordering on X . For each x ∈ X , denote (U )x = {Un(x)}n∈N. For each n ∈ N,let

Vn(x) = B(x, 1/n)◦,h(n, x) = Un(x) ∩ Vn(x),p(n, x) = min{y ∈ X : x ∈ h(n, y)},g(n, x) = Vn(x) ∩ (∩{h(i, p(i, x)) : i � n})

∩ (∩{U j (p(i, x)) : j, i � n, x ∈ U j (p(i, x))}),Gn = {g(n, x) : x ∈ X}.

12 1 The Origin of Generalized Metric Spaces

Then {Gn} is a development for X . Because otherwise, there exist x ∈ X and anopen neighborhood W of x satisfying the following condition: for each i ∈ N, thereis xi ∈ X such that x ∈ g(i, xi ) �⊂ W . Since x ∈ Vi (xi ), xi → x . Pick l, m ∈ Nsuch that B(x, 1/ l) ⊂ Um(x) ⊂ W . If x ∈ h(l, y) ⊂ Vl(y) for some y ∈ X , theny ∈ B(x, 1/ l) ⊂ Um(x), and hence p(l, x) ∈ Um(x). Thus, there is k ∈ N such thatUm(x) = Uk(p(l, x)). Since Uk(p(l, x)) ∩ h(l, p(l, x)) is an open neighborhood ofx , there is i0 ∈ N such that xi ∈ Uk(p(l, x))∩h(l, p(l, x))when i � i0, so p(l, xi ) �p(l, x) when i � i0. On the other hand, if i � l, then x ∈ g(i, xi ) ⊂ h(l, p(l, xi )),and hence p(l, x) � p(l, xi ). So p(l, xi ) = p(l, x) when i � max{i0, l}, it followsthat xi ∈ Uk(p(l, xi )). Hence g(i, xi ) ⊂ Uk(p(l, xi )) = Uk(p(l, x)) = Um(x) ⊂W when i � max{i0, l, k}, a contradiction. This contradiction shows that X is adevelopable space. �

1.3 Bases

The problem of seeking metrization theorems for spaces, i.e. looking for character-izations of metrizable spaces, is a general problem produced after proposing of theconcepts of metric spaces and topological spaces. This problem was solved perfectlyin the 1950s.

Definition 1.3.1 A family P in a space X is said to be locally finite at a pointx ∈ X [5], if there is a neighborhood V of x such that V meets at most finitely manyelements of P . If P is locally finite at every point x ∈ X , then P is said to belocally finite in X .

Bing, Nagata and Smirnov obtained their classic metrization theorem after theStone theorem.

Theorem 1.3.2 (The Bing–Nagata–Smirnov metrization theorem) The followingare equivalent for every regular space X:

(1) X is a metrizable space.(2) X has a σ -discrete base [62].(3) X has a σ -locally finite base [370, 428].

Proof (1) ⇒ (2). Suppose (X, d) is a metrizable space. For each n ∈ N, let Bn ={B(x, 1/2n)}x∈X . Then st(x,Bn) ⊂ B(x, 1/n). By the Stone theorem (see Theorem1.2.3), Bn has a σ -discrete open refinement Vn . It is easy to verify that

⋃n∈N Vn is

a σ -discrete base of X .(2) ⇒ (3) is obvious. We prove (3) ⇒ (1). Suppose X has a σ -locally finite base

B. For any open cover U of X , let V = {B ∈ B : there is U ∈ U such that B ⊂U }. Then V is a σ -locally finite refinement of U . So X is a paracompact space,hence a normal space. Denote B = ⋃n∈N Bn , where each Bn = {Bα : α ∈ Λn} islocally finite. For every n, m ∈ N and α ∈ Λn , let

Aα = ∪{A ∈ Bm : A ⊂ Bα}.

1.3 Bases 13

Then Aα ⊂ Bα . By the Urysohn lemma, there is a continuous function fα : X → Isuch that fα(Aα) ⊂ {1} and fα(X − Bα) ⊂ {0}. For every x, y ∈ X , define

dn,m(x, y) = min{1,

∑

α∈Λn| fα(x) − fα(y)|

}.

Then dn,m is a pseudo-distance on X . Rearrange {dn,m}n,m∈N as {dk}k∈N. Defineρ : X × X → R+ by

ρ(x, y) =∞∑

k=1

dk(x, y)

2k.

Then ρ is a distance on X and the metric topology generated by ρ is just the originaltopology for X . Thus X is a metrizable space. �

(1) ⇔ (2) and (1) ⇔ (3) of Theorem 1.3.2 are called the Bing metrizationtheorem and the Nagata-Smirnov metrization theorem respectively. By them we canobtain more metrization theorems.

Definition 1.3.3 ([63]) A familyP of subsets of X is said to be point-finite if (P)xis finite for every x ∈ X ; P is said to be compact-finite if (P)K is finite for everyK ∈ K (X).

Obviously, locally finite ⇒ compact-finite ⇒ point-finite.Corollary 1.3.4 ([63]) A space X is a metrizable space if and only if X is a regularspace with a σ -compact-finite base.

Proof We only need to prove the sufficiency. SupposeB is a σ -compact-finite basefor X . LetB = ⋃n∈N Bn , where eachBn is compact-finite. We first prove eachBnis locally finite. Otherwise, there is n ∈ N such that Bn is not locally finite at somepoint x ∈ X . Let (B)x = {Bi }i∈N and Pk = ⋂i�k Bi for each k ∈ N. Then (Bn)Pkis infinite for each k ∈ N, and hence there exist an infinite subset {Qk}k∈N ofBn anda sequence {xk} in X such that xk ∈ Pk ∩ Qk . Let F = {x} ∪ {xk : k ∈ N}. Then Fis a compact set in X , which contradicts the assumption that Bn is compact-finite.SoB is σ -locally finite base of X , and hence X is a metrizable space. �

Let X be a space and A ⊂ X . A familyP of X is call a neighborhood base of Ain X if, A ⊂ P◦ for each P ∈ P , and for each open set U in X containing A, thereexists P ∈ P such that P ⊂ U .Theorem 1.3.5 The following are equivalent for any space X:

(1) X is a metrizable space.(2) There is a sequence {Un} of open covers of X such that for each x ∈

X, {st2(x,Un)}n∈N is a neighborhood base of x in X [358].(3) There is a sequence {Un} of open covers of X such that for each K ∈

K (X), {st(K ,Un)}n∈N is a neighborhood base of K in X [215].

14 1 The Origin of Generalized Metric Spaces

(4) There is a development {Un} for X such that Un+1 is a star-refinement of Un[469].

(5) X is a collectionwise normal developable space [62] (the Bing metrization cri-terion).

Proof (1) ⇒ (2), (3) and (4). Suppose X is a metrizable space. For each n ∈ N, let

Un = {U ∈ τ(X) : diamU < 1/2n}.

Then {Un} satisfies conditions (2) and (3). Since X is a paracompact space, we cantakeH1 = U1 and letHn+1 be an open star-refinement ofUn+1 ∧ (∧i�n Hi ). Then{Hn} satisfies condition (4).

(4) ⇒ (2). It follows from the fact that if Un+1 is a star-refinement of Un , thenst2(x,Un+1) ⊂ st(x,Un).

(3)⇒ (2). We may assume thatUn+1 refinesUn and there exist x ∈ X and U ∈ τwith x ∈ U ∈ τ , such that, st2(x,Un) �⊂ U for each n ∈ N. Pick xn ∈ st2(x,Un)−U .Then there is yn ∈ st(x,Un) such that xn ∈ st(yn,Un), and hence yn → x . We mayassume yn ∈ U for each n ∈ N. Let K = {x} ∪ {yn : n ∈ N}. Then K ∈ K (X), sothere is m ∈ N such that st(K ,Um) ⊂ U . It follows that xm ∈ U , a contradiction.

(2) ⇒ (5). We may assume that Un+1 refines Un . Obviously, X is a developablespace. LetH = {Hα}α∈Λ be a discrete family of closed sets in X . For every α ∈ Λand x ∈ Hα , there is n(x) ∈ N such that st2(x,Un(x)) ⊂ X − ∪{Hβ : α �= β ∈ Λ}.Let Uα = ⋃x∈Hα st(x,Un(x)). Then Hα ⊂ Uα ∈ τ and {Uα}α∈Λ is a disjoint family.So X is a collectionwise normal space.

(5) ⇒ (1). Let {Un} be a development for X and X be a collectionwise normalspace. Since X is a subparacompact space, Un has a closed refinement

⋃m∈N Fn,m

such that Fn,m is a discrete family of closed sets in X . Since X is a collectionwisenormal space, we can take a discrete familyBn,m = {BF : F ∈ Fn,m} of open sets inX satisfying the following condition: for each F ∈ Fn,m , there is U ∈ Un such thatF ⊂ BF ⊂ U . Then ⋃n,m∈N Bn,m is a σ -discrete base of X . So X is a metrizablespace.

Definition 1.3.6 (1) A familyP in X is said to be closure-preserving at x ∈ X if,for any P ′ ⊂ P , x ∈ ∪P ′ whenever x ∈ ∪P ′. For a subset A of X , if P isclosure-preserving at every x ∈ A, thenP is said to be closure-preserving in A[326]. A regular space with a σ -closure-preserving base is called an M1-space[92].

(2) A family B in X is called a quasi-base for X if, for every x ∈ X and U ∈ τwith x ∈ U ∈ τ , there is B ∈ B such that x ∈ B◦ ⊂ B ⊂ U . A regular spacewith a σ -closure-preserving quasi-base is called an M2-space [92].

(3) A family P of ordered pairs of subsets of X is called a pair-base for X if Psatisfies the following conditions:

(i) (P1, P2) ∈ P ⇒ P1 ⊂ P2 and P1 ∈ τ ;(ii) for each x ∈ U ∈ τ , there is (P1, P2) ∈ P such that x ∈ P1 ⊂ P2 ⊂ U .

1.3 Bases 15

A family P of ordered pairs of sets in X is said to be cushioned in X [327] if,for everyP ′ ⊂ P , ∪{P1 : (P1, P2) ∈ P ′} ⊂ ∪{P2 : (P1, P2) ∈ P ′}. A space witha σ -cushioned base is called an M3-space [92].

A σ -cushioned base is usually called a σ -cushioned pair-base. Since a cushionedfamily is always a family of ordered pairs, so we call it a σ -cushioned base. In thisbook, we give the same treatment in all other places when we mention a cushionedfamily.

M1-spaces, M2-spaces and M3-spaces are unified called Mi -spaces [92]. It isobvious that metrizable ⇒ M1 ⇒ M2 ⇒ M3 ⇒ paracompact. Below we investigatethe properties of Mi -spaces. Obviously, Mi -spaces are additive.

Proposition 1.3.7 (1) M1-spaces are open hereditary.(2) M2-spaces and M3-spaces are hereditary.

Proof (1) Assume thatB is a σ -closure-preserving base for an M1-space X . If Z isan open subspace of X , then {B ∈ B : B ⊂ Z} is a σ -closure-preserving base forZ , so Z is also an M1-space.

(2) We prove it only for the M2-case. SupposeB is a σ -closure-preserving quasi-base for an M2-space X . If Z is a subspace of X , then B|Z a σ -closure-preservingquasi-base for Z , and hence Z is an M2-space. �

For familiesP1 and P2 of sets in spaces X and Y respectively, define

P1 × P2 = {P1 × P2 : P1 ∈ P1, P2 ∈ P2}.

For families P1 and P2 of ordered pairs of sets in spaces X and Y respectively,define

P1 × P2 = {(P1 × P2, Q1 × Q2) : (Pi , Qi ) ∈ Pi , i = 1, 2}.

We define∏

α∈Λ Pα in a similar way.

Lemma 1.3.8 If Pi (i = 1, 2) is a locally finite (resp. closure-preserving, cush-ioned) family for Xi , then P1 × P2 is a locally finite (resp. closure-preserving,cushioned) family for space X1 × X2.Proof We prove it only for the closure-preserving case.Let P1 = {Fα}α∈A and P2 = {Hβ}β∈B . We prove that for any C ⊂ A × B

∪{Fα × Hβ : (α, β) ∈ C} ⊂ ∪{Fα × Hβ : (α, β) ∈ C}.

Assume (x, y) ∈ X1 × X2 − ∪{Fα × Hβ : (α, β) ∈ C}. Let

U = X1 − ∪{Fα : x ∈ X1 − Fα, (α, β) ∈ C},V = X2 − ∪{Hβ : y ∈ X2 − Hβ, (α, β) ∈ C}.