Encyclopedia ofherfort/W_Hojka/EGT.pdf · Encyclopedia of General Topology Edited by Klaas Pieter...

Encyclopedia ofGeneral Topology

Encyclopedia of General Topology

Editors

Klaas Pieter HartFaculty of Electrical Engineering, Mathematics and Computer ScienceDelft University of TechnologyMekelweg 42628 CD Delft, The Netherlands

Jun-iti NagataUzumasa Higashiga-Oka 13-2Neyagawa-shiOsaka, 572-0841, Japan

Jerry E. VaughanDepartment of Mathematical SciencesUniversity of North Carolina at GreensboroP. O. Box 26170Greensboro, NC 27402-6170U.S.A.

Encyclopedia ofGeneral Topology

Edited by

Klaas Pieter Hart, Jun-iti Nagata and Jerry E. Vaughan

Associate Editors

Vitaly V. Fedorchuk, Gary Gruenhage, Heikki J.K. Junnila, Krystyna M. Kuperberg,Jan van Mill, Tsugunori Nogura, Haruto Ohta, Akihiro Okuyama,

Roman Pol, Stephen Watson

2004

Amsterdam – Boston – Heidelberg – London – New York – Oxford – ParisSan Diego – San Francisco – Singapore – Sydney – Tokyo

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Contents

Preface viiContributors ix

A Generalitiesa-01 Topological Spaces 3a-02 Modified Open and Closed Sets

(Semi-Open Set etc.) 8a-03 Cardinal Functions, Part I 11a-04 Cardinal Functions, Part II 15a-05 Convergence 18a-06 Several Topologies on One Set 22a-07 Comparison of Topologies (Minimal

and Maximal Topologies) 24

B Basic constructionsb-01 Subspaces (Hereditary (P)-Spaces) 31b-02 Relative Properties 33b-03 Product Spaces 37b-04 Quotient Spaces and Decompositions 43b-05 Adjunction Spaces 47b-06 Hyperspaces 49b-07 Cleavable (Splittable) Spaces 53b-08 Inverse Systems and Direct Systems 56b-09 Covering Properties 60b-10 Locally (P )-Spaces 65b-11 Rim(P)-Spaces 67b-12 Categorical Topology 74b-13 Special Spaces 76

C Maps and general types of spaces defined bymapsc-01 Continuous and Topological Mappings 83c-02 Open Maps 86c-03 Closed Maps 89c-04 Perfect Maps 92c-05 Cell-Like Maps 97c-06 Extensions of Maps 100c-07 Topological Embeddings

(Universal Spaces) 103c-08 Continuous Selections 107c-09 Multivalued Functions 110c-10 Applications of the Baire Category

Theorem to Real Analysis 113c-11 Absolute Retracts 117c-12 Extensors 122c-13 Generalized Continuities 126

c-14 Spaces of Functions in Pointwise Con-vergence 131

c-15 Radon–Nikodým Compacta 138c-16 Corson Compacta 140c-17 Rosenthal Compacta 142c-18 Eberlein Compacta 145c-19 Topological Entropy 147c-20 Function Spaces 150

D Fairly general propertiesd-01 The Low Separation Axioms T0 and T1 155d-02 Higher Separation Axioms 158d-03 Fréchet and Sequential Spaces 162d-04 Pseudoradial Spaces 165d-05 Compactness (Local Compactness,

Sigma-Compactness etc.) 169d-06 Countable Compactness 174d-07 Pseudocompact Spaces 177d-08 The Lindelöf Property 182d-09 Realcompactness 185d-10 k-Spaces 189d-11 Dyadic Compacta 192d-12 Paracompact Spaces 195d-13 Generalizations of Paracompactness 198d-14 Countable Paracompactness, Countable

Metacompactness, and RelatedConcepts 202

d-15 Extensions of Topological Spaces 204d-16 Remainders 208d-17 The Čech–Stone Compactification 210d-18 The Čech–Stone Compactifications

of N and R 213d-19 Wallman–Shanin Compactification 218d-20 H -Closed Spaces 221d-21 Connectedness 223d-22 Connectifications 227d-23 Special Constructions 229

E Spaces with richer structurese-01 Metric Spaces 235e-02 Classical Metrization Theorems 239e-03 Modern Metrization Theorems 242e-04 Special Metrics 247e-05 Completeness 251e-06 Baire Spaces 255e-07 Uniform Spaces, I 259e-08 Uniform Spaces, II 264

v

vi Contents

e-09 Quasi-Uniform Spaces 266e-10 Proximity Spaces 271e-11 Generalized Metric Spaces, Part I 273e-12 Generalized Metric Spaces, Part II 276e-13 Generalized Metric Spaces III:

Linearly Stratifiable Spaces andAnalogous Classes of Spaces 281

e-14 Monotone Normality 286e-15 Probabilistic Metric Spaces 288e-16 Approach Spaces 293

F Special propertiesf-01 Continuum Theory 299f-02 Continuum Theory (General) 304f-03 Dimension Theory (General Theory) 308f-04 Dimension of Metrizable Spaces 314f-05 Dimension Theory: Infinite Dimension 318f-06 Zero-Dimensional Spaces 323f-07 Linearly Ordered and Generalized

Ordered Spaces 326f-08 Unicoherence and Multicoherence 331f-09 Topological Characterizations

of Separable MetrizableZero-Dimensional Spaces 334

f-10 Topological Characterizations of Spaces 337f-11 Higher-Dimensional Local

Connectedness 341

G Special spacesg-01 Extremally Disconnected Spaces 345g-02 Scattered Spaces 350g-03 Dowker Spaces 354

H Connections with other structuresh-01 Topological Groups 359h-02 Topological Rings, Division Rings, Fields

and Lattices 365

h-03 Free Topological Groups 372h-04 Homogeneous Spaces 376h-05 Transformation Groups and Semigroups 379h-06 Topological Discrete Dynamical

Systems 385h-07 Fixed Point Theorems 402h-08 Topological Representations

of Algebraic Systems 409

J Influencies of other fieldsj-01 Descriptive Set Theory 417j-02 Consistency Results in Topology, I:

Quotable Principles 419j-03 Consistency Results in Topology, II:

Forcing and Large Cardinals 423j-04 Digital Topology 428j-05 Computer Science and Topology 433j-06 Non Standard Topology 436j-07 Topological Games 439j-08 Fuzzy Topological Spaces 443

K Connections with other fieldsk-01 Banach Spaces and Topology (I) 449k-02 Banach Spaces (and Topology) (II) 454k-03 Measure Theory, I 459k-04 Measure Theory, II 464k-05 Polyhedra and Complexes 470k-06 Homology 474k-07 Homotopy, I 480k-08 Homotopy, II 485k-09 Shape Theory 489k-10 Manifold 494k-11 Infinite-Dimensional Topology 497

Subject index 503

Preface

General Topology has experienced rapid growth during thepast fifty years, and nowadays its language and conceptspervade much of modern day mathematics. This book isintended for a broad community of scholars and studentsworking in mathematics and other areas who want to be-come acquainted quickly with the terminology and ideas ofGeneral Topology that may be relevant to their work. Thusthe book provides a source where the specialist and non-specialist alike can find short introductions to both the basictheory and the newest developments in General Topology.

Because the book is designed for the reader who wantsto get a general view of the terminology with minimal timeand effort there are very few proofs given; on occasion asketch of an argument will be given, more to illustrate a no-tion than to justify a claim. We assume that the reader hasa rudimentary knowledge of Set Theory, Algebra, Geometryand Analysis. A reader who wants to study the subject matterof one or more of the articles systematically (or who wantsto see the proof of a particular result) will find sufficient ref-erences at the end of each article as well as in the books inour list of standard references.

Guide to the reader

Titles of articles are given in the Table of Contents underthe following ten headings that roughly follow Section 54of the 2000 Mathematics Subject Classification as used byMathematical Reviews and Zentralblatt MATH.• A – Generalities• B – Basic constructions• C – Maps and general types of spaces defined by maps• D – Fairly general properties• E – Spaces with richer structures• F – Special properties• G – Special spaces• H – Connection with other structures• J – Influences of other fields• K – Connections with other fieldsTopological terms used in the encyclopedia are listed in

the Index. Terms defined within an article are indicated thus:compactness. Terms used in an article but defined elsewherewill be typeset thus: compactness (the first occurrence only);their definitions can be located by consulting the index.

There is a list of standard references, given below, that arecited uniformly by a letter system. Thus [E, 3.1] would referto the first section on compactness in Engelking’s GeneralTopology and [HvM, Chapter 18] to the chapter on compact

spaces in Recent Progress in General Topology (edited byHušek and van Mill).

Standard references

[E] R. Engelking, General Topology, 2nd edition, SigmaSer. Pure Math., Vol. 6, Heldermann, Berlin (1989).

[HvM] M. Hušek and J. van Mill, eds., Recent Progressin General Topology, North-Holland, Amsterdam(1992).

[Ke] J.L. Kelley, General Topology, Van Nostrand, NewYork (1955), Reprinted as: Graduate Texts in Math.,Vol. 27, Springer, New York (1975).

[Ku] K. Kunen, Set Theory. An Introduction to Indepen-dence Proofs, Studies Logic Found. Math., Vol. 102,North-Holland, Amsterdam (1980).

[KV] K. Kunen and J.E. Vaughan, eds., Handbook ofSet-Theoretic Topology, North-Holland, Amsterdam(1984).

[KI] K. Kuratowski, Topology, Vol. I, Academic Press,New York (1966).

[KII] K. Kuratowski, Topology, Vol. II, Academic Press,New York (1968).

[Kur] K. Kuratowski, Introduction to Set Theory and To-pology, PWN and Pergamon, Warsaw and Oxford(1977).

[vMR] J. van Mill and G.M. Reed, eds., Open Problems inTopology, North-Holland, Amsterdam (1990).

[MN] K. Morita and J. Nagata, eds., Topics in GeneralTopology, North-Holland Math. Library, Vol. 41,North-Holland, Amsterdam (1989).

[N] J. Nagata, Modern General Topology, 2nd new edi-tion, North-Holland Math. Library, Vol. 33, North-Holland, Amsterdam (1985).

Acknowledgments

We wish to thank Professor W. Moors for his help withthe English expression of some of the articles, ProfessorK. Yamada for his help in various ways, the Symposiumof General Topology of Japan for partial financial support,Professor A.V. Arhangel’skiı̆ for his help in organizing thearticles related to Cp(X)-theory, and Drs. A. Sevenster forhis encouragement throughout the project.

K.P. Hart, J.-I. Nagata and J.E. VaughanEditors

vii

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List of Contributors

Aarts, J.M., Delft, The Netherlands d19Aoki, Nobuo, Tokyo, Japan h06Arhangel’skiı̆, A.V., Athens, OH, USA c14, c16, c18Balcar, Bohuslav, Prague, Czech Republic f06Balogh, Zoltan T., Oxford, OH, USA g03Bella, Angelo, Catania, Italy d04Bennett, Harold, Lubbock, TX, USA f07Błaszczyk, Aleksander, Katowice, Poland d05Borges, Carlos R., Davis, CA, USA c09Brown, Ronald, Bangor, United Kingdom d10Bruckner, Andrew, Santa Barbara, CA, USA c10Brunet, Bernard, Aubiere, France j06Burke, Dennis K., Oxford, OH, USA c04, e12Cascales, B., Espinardo, Spain k01, k02Chaber, J., Warsaw, Poland e05, e06Charatonik, Janusz J., Wrocław, Poland and

Mexico D.F., Mexico f08Chiba, Keiko, Shizuoka, Japan b03Chigogidze, Alex, Saskatoon, SK, Canada c07Choban, M.M., Kishinev, Moldava c03Ciesielski, Krzysztof, Morgantown, WV, USA c13Coplakova, Eva, Delft, The Netherlands f06Daverman, Robert J., Knoxville, TN, USA c05, k10Dikranjan, Dikran, Udine, Italy h01Dobrowolski, Tadeusz, Pittsburg, KS, USA k11Dolecki, Szymon, Dijon, France a05Dow, Alan, Charlotte, NC, USA d16, d17, d18Dranishnikov, Alexander, Gainesville, FL, USA c12, k06Fedorchuk, Vitaly, Moscow, Russia d23García-Máynez, Adalberto, México D.F., Mexico e08Gartside, Paul, Pittsburgh, PA, USA e11Good, Chris, Birmingham, UK d08Hart, Klaas Pieter, Delft, The Netherlands

c20, d16, d17, d18, d23, e07, j02Hattori, Yasunao, Matsue, Japan e01, e04Heath, Robert W., Pittsburgh, PA, USA c06Hiraide, Koichi, Matsuyama, Japan h06Hirschorn, James, Helsinki, Finland k03, k04Hodel, Richard E., Durham, NC, USA e02, e03Hofmann, Karl Heinrichr, Darmstadt, Germany d01Hoshina, Takao, Tsukuba, Japan b10Hušek, Miroslav, Prague, Czech Republic b12Itō, Munehiko, Kochi, Japan j05Junnila, Heikki J.K., Helsinki, Finland d13Kawakami, Tomohiro, Wakayama, Japan h05

Kawamura, Kazuhiro, Tsukuba, Japan h07Kemoto, Nobuyuki, Oita, Japan d02Kopperman, Ralph, New York, NY, USA j04Koyama, A., Osaka, Japan k07, k08Künzi, Hans-Peter A., Rondebosch, South Africa e09Lowen, Robert, Antwerp, Belgium e16, j08Lutzer, David, Williamsburg, VA, USA f07Mancio-Toledo, Rubén, México D.F., Mexico e08Marciszewski, Witold, Warsaw, Poland c17, k11Mardešić, Sibe, Zagreb, Croatia k05, k09Mayer, John C., Birmingham, AL, USA f01McAuley, Louis F., Binghamton, NY, USA c02McCluskey, Aisling E., Galway, Ireland a07Michael, E., Seattle, WA, USA c08, d12Miller, Arnold W., Madison, WI, USA j01Mioduszewski, Jerzy, Katowice, Poland d21Misiurewicz, Michał, Indianapolis, IN, USA c19Mizokami, Takemi, Joetsu, Japan b05, b06Nagata, Jun-iti, Osaka, Japan c01Naimpally, Somashekhar, Toronto, Canada e10Namioka, I., Seattle, WA, USA c15, k01, k02Nogura, Tsugunori, Matsuyama, Japan a05Noiri, Takashi, Yatsushiro, Japan a02Nowak, Sławomir, Warsaw, Poland f11Nyikos, P.J., Columbia, SC, USA d14, e13Ohta, Haruto, Shizuoka, Japan b13Okuyama, Akihiro, Kobe, Japan b01Orihuela, J., Espinardo, Spain k01, k02Oversteegen, Lex G., Birmingham, AL, USA f01Pasynkov, Boris A., Moscow, Russia f03Pelant, Jan, Praha, Czech Republic d15Pol, E., Warsaw, Poland f04Pol, R., Warsaw, Poland e05, e06Porter, Jack, Lawrence, KS, USA d20Raja, M., Espinardo, Spain k01, k02Reilly, Ivan L., Auckland, New Zealand a06Rudin, Mary Ellen, Madison, WI, USA e14Sakai, Masami, Yokohama, Japan a01Scheepers, Marion, Boise, ID, USA j07Schommer, John J., Martin, TN, USA d09Sempi, Carlo, Lecce, Italy e15Shakhmatov, Dmitri, Matsuyama, Japan b07, h02Shapiro, Leonid B., Moscow, Russia d11Shimane, Norihito, Kagoshima, Japan b05, b06Simon, Petr, Prague, Czech Republic d03

ix

x

Soukup, Lajos, Budapest, Hungary g02Stephenson, R.M., Jr, Columbia, SC, USA d07Tall, Franklin D., Toronto, Canada j02, j03Tamano, Ken-ichi, Yokohama, Japan a03, a04Tanaka, Yoshio, Tokyo, Japan b04Terada, Toshiji, Yokohama, Japan h04Terasawa, Jun, Yokosuka, Japan g01Tironi, Gino, Trieste, Italy d04Toruńczyk, Henryk, Warszawa, Poland f05Trnková, Věra, Prague, Czech Republic h08Tuncali, H. Murat, North Bay, ON, Canada b11

Tymchatyn, E.D., Saskatoon, SK, Canada f02van Mill, Jan, Amsterdam, The Netherlands f09Vaughan, Jerry E., Greensboro, NC, USA d06Vermeer, Johannes, Delft, The Netherlands d20Watanabe, Tadashi, Yamaguchi, Japan b08West, James E., Ithaca, NY, USA c11, f10Wilson, Richard G., Mexico D.F., Mexico d22Yajima, Yukinobu, Yokohama, Japan b09Yamada, Kohzo, Shizuoka, Japan h03Yasui, Yoshikazu, Osaka, Japan b02Yung Kong, T., Flushing, NY, USA j04

A: Generalities

a-01 Topological spaces 3

a-02 Modified open and closed sets(semi-open set etc.) 8

a-03 Cardinal functions, part I 11

a-04 Cardinal functions, part II 15

a-05 Convergence 18

a-06 Several topologies on one set 22

a-07 Comparison of topologies (minimaland maximal topologies) 24

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a-1 Topological Spaces

The following is a brief survey of basic terminology used ingeneral topology. Engelking’s book [E] is one of the stan-dard texts on general topology. In the book [E, pp. 18–20]the readers can quickly review origins and background ofgeneral topology.

1. Topological spaces

Let X be a set and T be a family of subsets on X. If Tsatisfies the following conditions:

(O1) ∅ ∈ T and X ∈ T ,(O2) If U1 ∈ T and U2 ∈ T , then U1 ∩U2 ∈ T ,(O3) If T ′ ⊂ T , then ⋃T ′ ∈ T ,then T is called a topology on X and the pair (X,T ) iscalled a topological space (or space for short). Every el-ement of (X,T ) is called a point. Every member of T iscalled an open set of X or open in X. If {x} ∈ T , then thepoint x is called an isolated point of X. The complementof an open set is called a closed set of X or closed in X.If a set is open and closed in a topological space, then it iscalled open-and-closed or closed-and-open (or clopen forshort). Let C be the family of closed sets of a topologicalspace (X,T ). Then C satisfies the following conditions:(C3) ∅ ∈ C and X ∈ C ,(C3) If F1 ∈ C and F2 ∈ C , then F1 ∪ F2 ∈ C ,(C3) If C ′ ⊂ C , then ⋂C ′ ∈ C .

The intersection of countably many open sets of a topo-logical space need not be open. The intersection of count-ably many open sets is called a Gδ-set. The union of count-ably many closed sets of a topological space need not beclosed. The union of countably many closed sets is called anFσ -set. The complement of a Gδ-set is an Fσ -set. A topo-logical space in which every closed set is a Gδ-set is calleda perfect space. Countable unions of Gδ-sets are Gδσ -setsand their complements are Fσδ-sets. This procedure can becontinued to form the hierarchy of Borel sets.

Let (X,T ) be a topological space. A subfamily B of Tis called a base or basis for X (or even an open base) iffor every x ∈X and arbitrary U ∈ T containing the point xthere exists V ∈ B such that x ∈ V ⊂U , in other words everyopen set of X is the union of a subfamily of B. A topologicalspace X having a countable base is called second-countableor we say that X satisfies the second axiom of countability.A base B for X satisfies the following conditions:(B1) If U1,U2 ∈ B and x ∈ U1∩U2, then there exists U ∈ B

such that x ∈U ⊂U1 ∩U2,(B2) X =⋃B.

A subfamily S of T is called a subbase (or an open sub-base) for X if the family of all finite intersections of mem-bers in S is a base for X.

Let X be a topological space and x ∈A⊂X. We say thatA is a neighbourhood of x if there exists an open set U inX such that x ∈U ⊂A. Every open set containing a point xis, of course, a neighbourhood of x . Such a neighbourhoodis called an open neighbourhood of x . A family B(x) ofneighbourhoods of x ∈X is called a neighbourhood base ora local base at the point x if for every open set U containingx there exists V ∈ B(x) such that x ∈ V ⊂ U . A family is alocal subbase at a point if its finite intersections form a localbase at that point.

If every point of X has a countable neighbourhood base,thenX is called first-countable or we say thatX satisfies thefirst axiom of countability. Every second-countable spaceis first-countable. Let {B(x)}x∈X be a collection of neigh-bourhood bases of the points of X, which is called a neigh-bourhood system for X. It satisfies the following condi-tions:

(NB1) Every B(x) is non-empty and every member of B(x)contains x ,

(NB2) If U,V ∈ B(x), then there exists W ∈ B(x) such thatW ⊂U ∩ V ,

(NB3) If U ∈ B(x), then there exists a set V such that x ∈V ⊂ U and for every y ∈ V there exists W ∈ B(y)satisfying W ⊂ V .

Let A ⊂ B ⊂ X. We say that B is a neighbourhoodof a set A if there exists an open set U in X such thatA⊂ U ⊂ B . A neighbourhood base at A sometimes calledan outerneighbourhood base for A is a family B of neigh-bourhoods such that every neighbourhood of A contains amember of B.

Let A be a subset of a topological space X. We denote byIntA (intA or A◦) the union of all open sets of X containedin A. The set IntA is called the interior of A. It is the largestopen set of X contained in A. It is easy to check that a pointx belongs to IntA if and only if there exists a neighbourhoodU of x such that U ⊂A. The operator Int, called the interioroperator, satisfies the following conditions:

(IO1) IntX=X,(IO2) IntA⊂A,(IO3) Int(A∩B)= IntA∩ IntB ,(IO4) Int(IntA)= IntA.

We denote by Ā (clX A or ClX A) the intersection of allclosed sets of X containingA. The set Ā is called the closureof A. It is the smallest closed set of X containingA. It is easyto check that a point x belongs to Ā (is an adherent point

4 Section A: Generalities

of A) if and only if every neighbourhood of x intersects A.A point not in Ā is an exterior point of A.

The set Ā ∩X−A is called the boundary of A and de-noted by FrA (BrA, ∂A). The operator ·, called the closureoperator, satisfies the following conditions:

(CO1) ∅ = ∅,(CO2) A⊂ Ā,(CO3) A∪B = Ā∪B ,(CO4) (Ā)= Ā.

EXAMPLE 1. Let X= {a, b, c} and

T = {∅,X, {a, c}, {b, c}, {c}}.

Since T satisfies (O1), (O2) and (O3), (X,T ) is a topolo-gical space. The family of closed subsets of X is {∅,X, {b},{a}, {a, b}}. Obviously Int{a, b} = ∅ and clX{c} =X.

EXAMPLE 2. Let R be the set of real numbers. Let T bethe family of all subsets U of R satisfying the propertythat for each x ∈ U there exists an ε > 0 such that (x − ε,x + ε) ⊂ U . Since T satisfies (O1), (O2) and (O3), (R,T )is a topological space. The topology T is called the naturaltopology on R.

A subset U of a topological space X is called a regularopen set or an open domain if U = IntU holds. A subset Fof a topological space X is called a regular closed set or aclosed domain if F = IntA holds.

A point x of a topological space X is called an accumula-tion point or a cluster point of a set A⊂ X if x ∈ A− {x}holds, in other words every neighbourhood of x contains apoint of A besides x . The set of all accumulation points of Ais called the derived set of A, and denoted by Ad . A pointx of a topological space X is called a complete accumu-lation point of a set A ⊂ X if for every neighbourhood Uof x , |U ∩A| = |A| holds. If every neighbourhood of x con-tains uncountably many points of A, then the point is calleda condensation point of A.

A subset A of a topological space X is called a dense set(a nowhere dense set respectively) in X if Ā=X (Int Ā= ∅respectively) holds. If a topological space X has a countabledense subset, then X is called a separable space. Any count-able union of nowhere dense subsets is called a set of firstcategory or a meager set. Any set not of the first categoryis called a set of second category.

Let X be a topological space and A a family of subsetsof X. If X =⋃A holds, then A is called a cover or cov-ering of X, and a cover whose members are open (closedrespectively) in X is called an open cover (a closed coverrespectively) of X. If every point of X is contained in atmost finitely many (countably many respectively) membersof A, then A is called point-finite (point-countable respec-tively). If for every x ∈X there exists a neighbourhood of xwhich intersects at most one member (finitely many mem-bers respectively) of A, then A is called a discrete family

(a locally finite family respectively). If for every subfamilyA′ ⊂A,

⋃{Ā: A ∈A′}=

⋃A′

holds, then A is called closure-preserving. More strongly,if for every subfamily A′ = {Aα: α ∈ Γ } ⊂A and any Bα ⊂Aα (α ∈ Γ ),

⋃{Bα : α ∈ Γ

}=⋃{Bα : α ∈ Γ }

holds, then A is called hereditarily closure-preserving. IfA can be represented as a countable union of discrete fam-ilies, then A is called σ -discrete. In a similar way we de-fine σ -disjoint, σ -locally finite, σ -point-finite, σ -closure-preserving, σ -hereditarily closure-preserving, etc. Thefollowing implication holds.

discrete↓

locally finite↙ ↘

point-finite hereditarily closure-preserving↓ ↓

point-countable closure-preserving

There are several ways to give a set X a topology.Let B be a family of subsets of X satisfying the conditions

(B1) and (B2). Let

T ={U : U =

⋃B′ for some B′ ⊂ B

}.

Then T is a topology on X and B is a base for the topolo-gical space (X,T ). The topology T is called the topologygenerated by the base B. Let S be a family of subsets ofX satisfying (B2). Then the family B of all finite intersec-tions of members of S satisfies (B1) and (B2). The topologygenerated by B is called the topology generated by the sub-base S .

Let {B(x): x ∈X} be a collection of families of subsets ofa set X satisfying the conditions (NB1), (NB2) and (NB3).

T = {U : if x ∈ U, then x ∈ V ⊂Ufor some V ∈ B(x)}.

Then T is a topology on X and each B(x) is a neighbour-hood base of x in the topological space (X,T ). The topologyT is called the topology generated by the neighbourhoodsystem {B(x): x ∈X}.

Let X be a set and Int is an operator assigning to everyset A ⊂ X a set IntA ⊂ X ruled by (IO1), (IO2), (IO3)and (IO4). Let

T = {U : IntU =U}.


Then T is a topology on X and IntA is the interior of A inthe topological space (X,T ). The topology T is called thetopology generated by the interior operator Int.

Let X be a set and · is an operator assigning to everyset A⊂ X a set Ā⊂ X ruled by (CO1), (CO2), (CO3) and(CO4). Let

T = {X−A: Ā=A}.

Then T is a topology on X and Ā is the closure of A inthe topological space (X,T ). The topology T is called thetopology generated by the closure operator ·.

Let T1 and T2 be topologies on a set X. If T1 ⊂ T2, thenwe say that T2 is finer (stronger or larger) than T1, or T1is coarser (weaker or smaller) than T2. The finest topologyon X is the family of all subsets of X, and it is called thediscrete topology. A topological space with this topologyis called the discrete space. The coarsest topology on Xconsists of ∅ and X only, and it is called the anti-discretetopology or indiscrete topology. A topological space withthis topology is called the anti-discrete space or indiscretespace. All topologies on a set X are partially ordered by in-clusion ⊂.

Let (X,T ) be a topological space and Y a subset of X.Let

TY = {Y ∩U : U ∈ T }.

It is easy to check that TY satisfies the conditions (O1), (O2)and (O3), so it is a topology on Y . The topological space(Y,TY ) is called a subspace of X. The topology TY is calledthe induced topology or subspace topology. If Y is open(closed respectively) in X, then it is called an open sub-space (a closed subspace respectively) of X. Obviously aset A⊂ Y is open (closed respectively) in the subspace Y ifand only if there exists an open set U (a closed set F respec-tively) of X such that A= Y ∩U (A= Y ∩ F respectively).If a subspace Y of a topological space X does not have anyisolated points, then it is called dense-in-itself. A closed anddense in itself subspace is called a perfect set.

Let {Xα}α∈A be a family of topological spaces such thatXα ∩ Xβ = ∅, where A is an index set. For the union X =⋃

α∈AXα , let

T = {U : U ⊂X, U ∩Xα is open in Xαfor any α ∈A}.

It is easy to check that T satisfies the conditions (O1), (O2)and (O3), so it is a topology on X. The topological space(X,T ) is called the topological sum (also discrete sum) ofthe spaces {Xα}α∈A and denoted by ⊕α∈AXα .

Let {Xα}α∈A be a family of topological spaces. For theCartesian product X =∏α∈AXα , let

B = {∏α∈AUα : Uα is open in Xα, Uα �=Xαonly for finitely many α ∈A}.

It is easy to check that B satisfies the conditions (B1) and(B2), so it generates a topology T on X. The topologicalspace (X,T ) is called the product space of the spaces{Xα}α∈A and the topology T is called the Tychonoff prod-uct topology on X. The base B is called the canonical basefor X.

2. Networks, k-networks and weak bases

We recall the notions of networks, k-networks and weakbases of topological spaces. They behave like bases, butevery member of such families need not be open. Thereforesuch families are easier to deal with than bases.

A family N of subsets of a topological space X is called anetwork for X if for every x ∈X and any neighbourhoodUof x there exists N ∈N such that x ∈N ⊂ U . A base for Xis a network. The family {{x}: x ∈X} is, for instance, a net-work for X. The notion of a network was introduced in [3] toinvestigate metrization conditions of compact spaces. It wasproved in [3] that the network weight and the weight areequal for compact spaces. Thus every compact space witha countable network is second-countable, hence metrizableby Urysohn’s theorem [E, 4.2.9]. Michael named a regularspace having a countable network as a cosmic space [16].It is known that a regular space X is the image of a sepa-rable metric space under a continuous map if and only ifX is cosmic, see [16]. A regular space with a σ -locally fi-nite network is called a σ -space, which was introduced byOkuyama [18]. The class of σ -spaces is important in the the-ory of generalized metric spaces.

A family P of subsets of a topological space X is calleda k-network (pseudobase respectively) for X if wheneverC ⊂ U with C compact and U open in X, there exists P ′of finitely many members of P (P ∈ P respectively) suchthat C ⊂⋃P ′ ⊂ U (C ⊂ P ⊂ U respectively). The notionsof a k-network and a pseudobase are due to [20] and [16]respectively. Obviously a pseudobase is a k-network, and ak-network is a network. According to [16], a regular spacehaving a countable pseudobase is called an ℵ0-space. If Pis a countable k-network for a topological space X, then thefamily of the unions of finitely many members of P is ob-viously a pseudobase for X. Hence we may say that an ℵ0-space is a topological space having a countable k-network.Since every separable metric space is second-countable, itis an ℵ0-space. Conversely every first-countable ℵ0-space isseparable and metrizable [16]. According to [20], a regularspace with a σ -locally finite k-network is called an ℵ-space.Since every metric space has a σ -locally finite base, it is anℵ-space. Conversely every first-countable ℵ-space is metriz-able [19].

As represented by Nagata–Smirnov’s metrization theo-rem, metrizability of topological spaces can be characterizedby means of the existence of special bases. Similarly we canmake use of a k-network to give inner characterizations ofimages of metric spaces by special continuous maps (e.g.,


quotient maps, closed maps, etc). We recall two typical re-sults in this direction. Michael proved in [16] that a regularspace X is the image of a separable metric space by a quo-tient map if and only if X is a k-space and an ℵ0-space.Foged proved in [12] that a regular space X is the image ofa metric space by a closed map if and only if X is a Fréchetspace with a σ -hereditarily closure-preserving k-network.

The notion of a k-network is useful in investigating func-tion spaces. For topological spaces X and Y , let C(X,Y ) bethe function space of all continuous maps from X to Y withthe compact-open topology. Michael proved in [16] that ifX and Y are ℵ0-spaces, then C(X,Y ) is also an ℵ0-space.Foged proved in [11] that if X is an ℵ0-space and Y is anℵ-space, then C(X,Y ) is a paracompact ℵ-space.

The field of the theory of a k-network is extensive. It isclosely related to other branches in general topology. Reg-ular spaces with a point-countable k-network were stud-ied in detail in [13]. Shibakov proved in [22] that a Haus-dorff sequential topological group with a point-countablek-network is metrizable if its sequential order is less thanω1. Further properties of sequential topological groups witha point-countable k-network were studied in [15]. The read-ers can find many papers on k-networks in the references of[24] and [25].

Let X be a topological space. For every x ∈ X let Txbe a family of subsets of X containing x . If the collection{Tx : x ∈X} satisfies(1) for every x ∈X the intersections of finitely many mem-

bers of Tx belong to Tx and(2) U ⊂ X is open in X if and only if x ∈ U implies x ∈

T ⊂U for some T ∈ Tx ,then it is called a weak base for X and each individual Txa weak neighbourhood base. This notion was introducedby Arhangel′skiı̆ [6, p. 129] to study symmetrizable spaces.A topological space X is said to satisfy the weak first axiomof countability or X is weakly first-countable if it has aweak base {Tx : x ∈X} such that each Tx is countable. Everysymmetrizable space is weakly first-countable and a topo-logical space is first-countable if and only if it is a weaklyfirst-countable Fréchet space [6]. A regular space with a σ -locally finite weak base is called a g-metrizable space [23].Foged proved in [10] that a topological space is g-metrizableif and only if it is a weakly first-countable ℵ-space.

3. Special bases

Since Alexandroff–Urysohn’smetrization theorem [E, 5.4.9],many special bases related to developable spaces have beenstudied.

A base B for a topological space X is a uniform baseor a point-regular base if for every point x ∈ X and everyneighbourhoodU of x the set

{B ∈ B: x ∈B, B ∩ (X−U) �= ∅}

is finite. It is easy to check that a base B for a topologicalspace X is uniform if and only if for every x ∈ X, everyfamily of countably infinite members of B containing x isa neighbourhood base at the point x . The notion of a uni-form base is due to Alexandroff [1] and he proved in thepaper that a topological space is metrizable if and only if itis collectionwise normal and has a uniform base. Topolo-gical spaces having a uniform base are closely related to de-velopable spaces. Heath proved in [14] that a T2-space hasa uniform base if and only if it is metacompact and devel-opable. Topological spaces having a uniform base were char-acterized as the images of metric spaces under continuouscompact open maps [6].

Arhangel’skiı̆ introduced in [4] a stronger version of a uni-form base. A base B for a topological space X is called aregular base if for every point x ∈X and every neighbour-hood U of x there exists a neighbourhood V of x such thatV ⊂U and the set

{B ∈ B: B ∩ V �= ∅ and B ∩ (X−U) �= ∅}

is finite. He proved in that paper that a topological space ismetrizable if and only if it is a T1-space and has a regularbase.

A base B for a topological space X is called a base ofcountable order if for every point x ∈X and any decreasingfamily {Bn: n ∈ ω} of distinct members of B containing x thefamily is a neighbourhood base at the point x . Obviously auniform base is a base of countable order. This notion was in-troduced by Arhangel’skiı̆ in [5]. He proved in [5] that everyregular developable space has a base of countable order, andthat a paracompact T2-space with a base of countable orderis metrizable. Worrell and Wicke gave a characterization ofdevelopable spaces in terms of bases of countable order. In-deed they proved in [26] that a T1-space is developable ifand only if it is submetacompact and has a base of count-able order. Hence, by Bing’s metrization theorem [E, 5.4.1],a collectionwise normal space is metrizable if and only if itis submetacompact and has a base of countable order [26].

For a family A of subsets of a set X and x ∈ X weput ord(x,A) = |{A ∈ A: x ∈ A}|. A family B =⋃n∈ω Bnof open subsets of a topological space X is called a θ -base (δθ -base respectively) if for every point x ∈ X andany neighbourhood U of x there exist n ∈ ω and B ∈Bn such that x ∈ B ⊂ U and 1 � ord(x,Bn) < ω (1 �ord(x,Bn) � ω respectively). Every σ -point finite base is aθ -base. A δθ -base is a common generalization of a θ -baseand a point-countable base. The notion of a θ -base is due toWorrell and Wicke [26] and they proved in this paper thata topological space X is developable if and only if X hasa θ -base and every closed subset of X is a Gδ-set. Hence,by Bing’s metrization theorem [E, 5.4.1], a collectionwisenormal space is metrizable if and only if it has a θ -baseand every closed subset is a Gδ-set [26]. Later Bennett andLutzer proved in [8] that a regular space has a θ -base if andonly if it is quasi-developable. The notion of a δθ -base wasintroduced by Aull [7]. Aull proved in [7] that a topological


space has a θ -base if and only if it is a weak σ -space andhas a δθ -base, where a topological space X is called a weakσ -space if it has a σ -disjoint network such that each disjointfamily is discrete in its union. Chaber proved in [9] that asubmetacompact β-space with a δθ -base is developable. Inparticular, by Bing’s metrization theorem [E, 5.4.1], everycompact T2-space with a δθ -base is metrizable .

Point-countable bases were introduced by Alexandroffand Urysohn [2]. They proved in [2] that a locally separa-ble space with a point-countable base is metrizable. Pono-marev proved in [21] that a T1-space is an image of a met-ric space by a continuous open s-map if and only if it hasa point-countable base. Miščenko proved in [17] that everycountably compact T1-space with a point-countable base ismetrizable.

References

[1] P. Alexandroff, On the metrization of topologicalspaces, Bull. Acad. Pol. Sci. Sér. Math. 8 (1960), 135–140 (in Russian).

[2] P. Alexandroff and P. Urysohn, Mémoire sur lesespaces topologiques compacts, Verh. Akad. Weten-sch. Amsterdam 14 (1929), 1–96.

[3] A.V. Arhangel’skiı̆, An addition theorem for the weightof sets lying in bicompcta, Dokl. Akad. Nauk. SSSR126 (1959), 239–241.

[4] A.V. Arhangel’skiı̆, On the metrization of topologicalspaces, Bull. Acad. Pol. Sci. Sér. Math. 8 (1960), 589–595 (in Russian).

[5] A.V. Arhangel’skiı̆, Certain metrization theorems, Usp.Math. Nauk 18 (1963), 139–145 (in Russian).

[6] A.V. Arhangel’skiı̆, Mappings and spaces, RussianMath. Surveys 21 (1966), 115–162.

[7] C. Aull, Quasi-developments and δθ -bases, J. LondonMath. Soc. 9 (1974), 197–204.

[8] H. Bennett and D. Lutzer, A note on weak θ -refi-nability, Gen. Topology Appl. 2 (1972), 49–54.

[9] J. Chaber, On point-countable collections and mono-tonic properties, Fund. Math. 94 (1977), 209–219.

[10] L. Foged, On g-metrizability, Pacific J. Math. 98(1982), 327–332.

[11] L. Foged, Characterizations of ℵ-spaces, Pacific J.Math. 110 (1984), 59–63.

[12] L. Foged, Characterization of closed images of metricspaces, Proc. Amer. Math. Soc. 95 (1985), 487–490.

[13] G. Gruenhage, E. Michael and Y. Tanaka, Spaces deter-mined by point-countable covers, Pacific J. Math. 113(1984), 303–332.

[14] R.W. Heath, Screenability, pointwise paracompactnessand metrization of Moore spaces, Canad. J. Math. 16(1964), 763–770.

[15] C. Liu, M. Sakai and Y. Tanaka, Topological groupswith a certain point-countable cover, Topology Appl.119 (2002), 209–217.

[16] E. Michael, ℵ0-spaces, J. Math. Mech. 15 (1966), 983–1002.

[17] A. Miščenko, Spaces with a point-countable base, So-viet Math. Dokl. 3 (1962), 855–858.

[18] A. Okuyama, Some generalizations of metric spaces,their metrization theorems and product spaces, Sci.Rep. Tokyo Kyoiku Daigaku Sec. A 9 (1967), 236–254.

[19] P. O’Meara, A metrization theorem, Math. Nachr. 45(1970), 69–72.

[20] P. O’Meara, On paracompactness in function spaceswith the compact-open topology, Proc. Amer. Math.Soc. 29 (1971), 183–189.

[21] V.I. Ponomarev, Axioms of countability and continu-ous maps, Bull. Acad. Pol. Sci. 8 (1960), 127–134 (inRussian).

[22] A. Shibakov, Metrizability of sequential topologicalgroups with point-countable k-networks, Proc. Amer.Math. 126 (1998), 943–947.

[23] F. Siwiec, On defining a space by a weak base, PacificJ. Math. 52 (1974), 233–245.

[24] Y. Tanaka, Theory of k-networks, Questions AnswersGen. Topology 12 (1994), 139–164.

[25] Y. Tanaka, Theory of k-networks II, Questions AnswersGen. Topology 19 (2001), 27–46.

[26] J.M. Worrell and H.H. Wicke, Characterizations ofdevelopable topological spaces, Canad. J. Math. 17(1965), 820–830.

Masami SakaiYokohama, Japan


a-2 Modified Open and Closed Sets(Semi-Open Set etc.)

1. θ -closed and δ-closed sets

A subset A of a topological space (X, τ) is called reg-ular open if A = Int(Cl(A)). The complement of a reg-ular open set is called regular closed. A point x ∈ X iscalled a θ -cluster (δ-cluster) point of A if A ∩ Cl(U) �= ∅(A ∩ Int(Cl(U)) �= ∅) for every open set U of X contain-ing x . The set of all θ -cluster (δ-cluster) points of A iscalled the θ -closure (δ-closure) and is denoted by Clθ (A)(Clδ(A)). A subset A is called θ -closed (δ-closed) [14] ifClθ (A) = A (Clδ(A) = A). The complement of a θ -closed(δ-closed) set is called θ -open (δ-open). The family of reg-ular open sets of (X, τ) is not a topology. But it is a basefor a topology τs called the semiregularization of τ . Ifτs = τ , then (X, τ) is called semiregular. For any sub-set S ⊂ X, Cl(S) ⊂ Clδ(S) ⊂ Clθ (S) and for any U ∈ τCl(U)= Clδ(U)= Clθ (U) [14].

The following are equivalent: (a) (X, τ) is Hausdorff;(b) for any distinct points x, y ∈X Clθ ({x})∩Clθ ({y})= ∅;(c) every compact set of X is θ -closed. A space (X, τ) iscalled almost regular if for any regular closed set F and apoint x /∈ F , there exist disjoint U,V ∈ τ such that F ⊂ Uand x ∈ V . The following are equivalent: (a) (X, τ) is regu-lar; (b) Clθ (A)= Cl(A) for every subset A⊂ X; (c) (X, τ)is almost-regular and semiregular. The following are equiv-alent: (a) (X, τ) is almost-regular; (b) Clθ (A)= Clδ(A) forevery subset A ⊂ X; (c) (X, τs) is regular; (d) Clθ (A) = Afor every regular closed set A of X.

2. Semi-open sets

A subset A of (X, τ) is semi-open [9] if A ⊂ Cl(Int(A)).The family SO(X, τ) of semi-open sets of (X, τ) is not a to-pology. The complement of a semi-open set is called semi-closed. A semi-open and semi-closed set is called semireg-ular [4] or regular semi-open. The semi-closure sCl(A) ofA is defined as

sCl(A)=⋂{

F : A⊂ F, X \F ∈ SO(X, τ)}.

If U ∈ SO(X, τ), then sCl(U) is semiregular. A subset Aof (X, τ) is called semi-θ -open if for each x ∈ A there ex-ists U ∈ SO(X, τ) such that x ∈ U ⊂ sCl(U) ⊂ A. By us-ing semi-open sets, many separation axioms, covering prop-erties and functions have been introduced and investigated.For example, we can mention semi-Hausdorff, semiregular,s-compact (every semi-open cover has a finite subcover),

semi-continuous, irresolute, etc. A function f :X → Y iscalled semi-continuous or quasi-continuous (respectivelyirresolute) if f−1(V ) is semi-open in X for every open (re-spectively semi-open) set V of Y .

A space (X, τ) is called S-closed [13] (s-closed [4]) iffor every cover {Vα: α ∈ Δ} of X by semi-open sets thereexists a finite subset Δ0 of Δ such that X =⋃α∈Δ0 Cl(Vα)(X=⋃α∈Δ0 sCl(Vα)). Every s-closed space is S-closed andevery S-closed space is quasi H -closed (for every opencover {Vα: α ∈ Δ} of X there exists a finite subset Δ0 ofΔ such that X = ⋃α∈Δ0 Cl(Vα)). For a Hausdorff space(X, τ), the following are equivalent: (a) (X, τ) is s-closed;(b) (X, τ) is S-closed; (c) (X, τ) is quasi H -closed ex-tremally disconnected. For a space (X, τ), the following areequivalent: (a) (X, τ) is S-closed; (b) every regular closedcover of X has a finite subcover; (c) every proper regularopen set of (X, τ) is an S-closed subspace. Compactness andS-closedness are independent of each other. S-closednessdoes not behave like compactness. The following are typicalproperties: a closed set of an S-closed space is not always S-closed; the product space of two S-closed spaces need notbe S-closed; S-closedness need not be preserved by continu-ous surjections; the inverse images of S-closed spaces underperfect maps are not necessarily S-closed.

For a space (X, τ), the following are equivalent: (a) (X, τ)is s-closed; (b) every cover of X by semiregular sets has afinite subcover; (c) every cover of X by semi-θ -open setshas a finite subcover. Moreover, s-closedness has the follow-ing properties: s-closedness is preserved by open continuoussurjections and if the product space

∏α∈ΔXα is s-closed

then each space Xα is s-closed but the converse is not true.

3. Preopen sets

A subset A of (X, τ) is called preopen [11], locally denseor nearly open if A ⊂ Int(Cl(A)). The complement of apreopen set is called preclosed. The family PO(X, τ) ofpreopen sets of (X, τ) is not a topology. For a subset Aof (X, τ), the following are equivalent: (a) A ∈ PO(X, τ);(b) A is the intersection of an open set and a dense set;(c) A is a dense subset of some open subspace; (d) sCl(A)=Int(Cl(A)). Here are basic properties of preopen sets: (1)every singleton is either preopen or nowhere dense; (2) thearbitrary union of preopen sets is preopen; (3) the finite inter-section of preopen sets need not be preopen; (4) preopennessand semi-opennes are independent of each other; (5) a subsetA is regular open if and only if it is semi-closed and preopen.

a-2 Modified open and closed sets (semi-open set etc.) 9

Extremally disconnected spaces are characterized by pre-open and semi-open sets. For a space (X, τ), the follow-ing are equivalent: (a) (X, τ) is extremally disconnected;(b) every regular closed subset of X is preopen; (c) everysemi-open set of X is preopen; (d) the closure of every pre-open set is open.

A space (X, τ) is called hyperconnected, irreducible ora D-space if every nonempty open set of X is dense. Fora space (X, τ), the following are equivalent: (a) (X, τ) ishyperconnected; (b) sCl(A) = X for every nonempty pre-open set A; (c) every nonempty preopen set of X is dense;(d) every nonempty semi-open set is dense; (e) X is not ex-pressed as the union of two disjoint nonempty semi-opensets of X.

By using preopen sets, many separation axioms and func-tions are defined and investigated. However, we deal withonly important covering properties: a space (X, τ) is astrongly compact space (or a p-closed space [7]) if forevery cover {Vα: α ∈ Δ} of X by preopen sets there ex-ists a finite subset Δ0 of Δ such that X =⋃α∈Δ0 Vα (X =⋃

α∈Δ0 pCl(Vα)). Every strongly compact space is p-closedand every p-closed space is quasi H -closed. If (X, τ) isp-closed and extremally disconnected, then it is s-closed.

4. α-open sets

A subset A of (X, τ) is called α-open [12] if A ⊂Int(Cl(Int(A))). The family τα of α-open sets of (X, τ)is a topology which is finer than τ . The complementof an α-open set is called α-closed. For a subset A of(X, τ), the following are equivalent: (a) A ∈ τα ; (b) A issemi-open and preopen; (c) A ∩ B ∈ SO(X, τ) for everyB ∈ SO(X, τ); (d) A = O \ N , where O ∈ τ and N isnowhere dense; (e) there exists U ∈ τ such that U ⊂ A ⊂sCl(U) = Int(Cl(U)). Here are some fundamental prop-erties of α-open sets: (1) the intersection of semi-open(preopen) set and an α-open set is semi-open (preopen),(2) SO(X, τ)= SO(X, τα) and PO(X, τ)= PO(X, τα), (3)α-open sets A,B are disjoint if and only if Int(Cl(Int(A)))and Int(Cl(Int(B))) are disjoint. As a consequence of (3) wecan obtain new characterizations of Hausdorffness, regular-ity, normality and connectedness by replacing open sets withα-open sets.

5. b-open and β-open sets

A subset A of (X, τ) is called b-open [3] ifA⊂ Int(Cl(A))∪Cl(Int(A)). The complement of a b-open set is calledb-closed. The family BO(X, τ) of b-open sets of (X, τ) isnot a topology. Every semi-open (preopen) set is b-open butthe converses are not true. There are some basic properties:(1) the union of any family of b-open sets is b-open, (2) theintersection of a b-open set and an α-open set is b-open,(3) BO(X, τ)= BO(X, τα).

A subset A of (X, τ) is called β-open [1] or semi-preopen [2] if A ⊂ Cl(Int(Cl(A))). The complement ofa β-open (semi-preopen) set is called β-closed (semi-preclosed). The family SPO(X, τ) of semi-preopen setsof (X, τ) is not a topology. For a subset A of (X, τ), thefollowing are equivalent: (a) A ∈ SPO(X, τ); (b) Cl(A)is regular closed; (c) there exists U ∈ PO(X, τ) such thatU ⊂ A⊂ Cl(U); (d) sCl(A) ∈ SO(X, τ); (e) A= S ∩D forS ∈ SO(X, τ) and a dense set D. There are some basic prop-erties of semi-preopen sets: (1) the arbitrary union of semi-preopen sets is semi-preopen, (2) BO(X, τ) ⊂ SPO(X, τ),(3) SPO(X, τ) = SPO(X, τα), (4) the intersection of anα-open set and a semi-preopen set is semi-preopen.

A space (X, τ) is called β-compact if every cover of Xby β-open sets has a finite subcover. However, it was shownin [8] that infinite β-compact spaces do not exist.

The following relations hold among modifications of opensets stated above.

regular open → semiregular↓ ↓

θ -open → δ-open → semi-open → b-open→ β-open↓ ↑ ↑

open → α-open → preopen

6. Generalized closed sets

In 1970, Levine [10] introduced the concept of generalizedclosed sets as a generalization of closed sets in topologicalspaces. Recently, the study of modifications of generalizedclosed sets has found considerable interest among generaltopologists. One of the reasons is that these concepts arequite natural. The following is the original definition: A sub-set A of (X, τ) is called generalized closed (g-closed) ifCl(A) ⊂ U whenever A ⊂ U and U ∈ τ . There are manymodifications of g-closed sets.

By replacing Cl(A) with Clδ(A) and Clθ (A), δ-gene-ralized closed [5] sets and θ -generalized closed [6] setsare obtained. These sets play an important role in studyof the digital line. The digital line (Z, κ) is the set Zof all integers equipped with the topology κ generated by{{2n−1,2n,2n+1}:n ∈ Z}. A space (X, τ) is called a T3/4-space [5] if every δ-generalized closed set of X is δ-closed.A T3/4-space places strictly between a T1-space and a T1/2-space (every g-closed set is closed). The Khalimsky line orso called digital line is a T3/4-space but it is not T1.

References

[1] M.E. Abd El-Monsef, S.N. El-Deeb and R.A. Mah-moud, β-open sets and β-continuous mappings, Bull.Fac. Sci. Assiut Univ. 12 (1983), 77–90.

[2] D. Andrijević, Semi-preopen sets, Mat. Vesnik 38(1986), 24–32.


[3] D. Andrijević, On b-open sets, Mat. Vesnik 48 (1996),59–64.

[4] G. Di Maio and T. Noiri, On s-closed spaces, Indian J.Pure Appl. Math. 18 (1987), 226–233.

[5] J. Dontchev and M. Ganster, On δ-generalized closedsets and T3/4-spaces, Mem. Fac. Sci. Kochi Univ. Ser.A Math. 17 (1996), 15–31.

[6] J. Dontchev and H. Maki, Groups of θ -generalizedhomeomorphisms and the digital line, Topology Appl.83 (1999), 113–128.

[7] J. Dontchev, M. Ganster and T. Noiri, On p-closedspaces, Internat. J. Math. Math. Sci. 24 (2000), 203–212.

[8] M. Ganster, Some remarks on strongly compact spacesand semi compact spaces, Bull. Malaysian Math. Soc.(2) 10 (1987), 67–70.

[9] N. Levine, Semi-open sets and semi-continuity in topo-

logical spaces, Amer. Math. Monthly 70 (1963), 36–41.

[10] N. Levine, Generalized closed sets in topology, Rend.Circ. Mat. Palermo (2) 19 (1970), 89–96.

[11] A.S. Mashhour, M.E. Abd El-Monsef and S.N. El-Deeb, On precontinuous and weak precontinuous map-pings, Proc. Math. Phys. Soc. Egypt 53 (1982), 47–53.

[12] O. Njåstad, On some classes of nearly open sets, PacificJ. Math. 15 (1965), 961–970.

[13] T. Thompson, S-closed spaces, Proc. Amer. Math. Soc.60 (1976), 335–338.

[14] N.V. Veličko, H -closed topological spaces, Amer.Math. Soc. Transl. (2) 78 (1968), 103–118.

Takashi NoiriYatsushiro, Japan

a-3 Cardinal functions, Part I 11

a-3 Cardinal Functions, Part I

A function ϕ which assigns a cardinal ϕ(X) to each topo-logical space X is called a cardinal function or a cardi-nal invariant if it is a topological invariant, i.e., if we haveϕ(X)= ϕ(Y ) whenever X and Y are homeomorphic.

Here we assume that the values of cardinal functions arealways infinite cardinals. The simplicity of infinite cardinalarithmetic (i.e., κ + λ = κ · λ = λ whenever κ and λ areinfinite cardinal numbers with κ � λ) helps us to simplifyour statements.

We adopt the following set-theoretic notation: cardinalnumbers are initial ordinals, i.e., κ is a cardinal if and onlyif it is the smallest ordinal of the cardinality |κ |. κ,λ, . . .always denote cardinal numbers.ω and ω1 are used to denotethe first infinite ordinal (cardinal) and the first uncountableordinal (cardinal) respectively. So, for each cardinal κ , wehave κ + ω = κ if κ is infinite, and κ +ω = ω if κ is finite.To avoid typesetting problems, 2κ is often denoted by expκ .Thus expω = 2ω is the cardinality c of the continuum, inother words, the cardinality of the set of real numbers.

Let (X, τ) be a topological space. Obviously, the cardi-nality |X| of X is the simplest cardinal function. Recallthat a subfamily B of τ is a base if every member of τ isthe union of a subfamily of B. The most important cardi-nal function is the weight w(X) of X, which is defined byw(X)=min{|B|: B is a base for X}+ω. Here ω is added tomake the value infinite. A space X satisfies the second ax-iom of countability, or is second-countable if w(X) � ω,in other words, if it has a countable base. In the early stage ofgeneral topology, first, Euclidean spaces and its subspaceswere studied. In the following stage, separable metrizablespaces played an important role. Indeed, for example, Kura-towski mainly concentrated on separable metrizable spacesin his book [4]. Kinds of countability played an importantrole in the classical theory. For a regular T1-space X, thefollowing are equivalent:

(a) X is separable metrizable,(b) w(X)= ω (i.e., X is second-countable),(c) X can be embedded in the product Iω of countably

many copies of the unit interval I = [0,1].More generally, for a completely regular T1-space X and aninfinite cardinal κ , the following (b′) and (c′) are equivalent:(b′) w(X) = κ , (c′) X can be embedded in the product Iκof κ many copies of the unit interval I = [0,1], which iscalled Tychonoff cube of weight κ . Hence weight is a car-dinal function which measures, in some sense, the size of atypical space in which the space can be embedded.

In Cardinal functions Part I, we give a brief introduc-tion to simple and well-known cardinal functions, which areobtained as generalizations of classical countability axioms

(for more information see [3], [KV, Chapter 1], [E, Prob-lems, Cardinal functions I, II, III, IV]). The inequalities be-tween the cardinal functions defined here are summarized in[KV, Chapter 1, §3, Figure 1].

Now we turn our attention to other cardinal invariants.For a metrizable space X, the separability of X can becharacterized by some other cardinal functions. Let X bea topological space. The Lindelöf degree (Lindelöf num-ber) L(X) (sometimes denoted by l(X)) of X is defined byL(X) = min{κ : every open cover of X has a subcover ofcardinality � κ} + ω. If L(X) = ω, i.e., every open coverhas a countable subcover, we say that X is Lindelöf. De-fine the density d(X) of X by d(X) = min{|D|: D is adense subset of X}. If d(X) = ω, we say that X is a sepa-rable space. Define the cellularity (Souslin number) c(X)of X by c(X) = min{κ : every family of pairwise disjointnonempty open subsets of X has cardinality � κ} + ω. Ifc(X)= ω, we say that X has the countable chain condition(Souslin property). Countable chain condition is abbrevi-ated as ccc. The spread s(X) and the extent e(X) are de-fined as follows: s(X) = sup{|D|: D is a discrete subset ofX} + ω, and e(X) = sup{|D|: D is a discrete closed subsetof X} +ω.

For a metrizable space X, we have

w(X)= L(X)= d(X)= c(X)= s(X)= e(X).For a completely metrizable space X, we have either |X| =w(X) or |X| =w(X)ω (all these results can be found in [KV,Chapter 1, §8]).

Those cardinal functions are not equal for general topo-logical spaces. However there are some relations betweenthem. For example, we have the following inequalities:L(X)�w(X), c(X)� d(X)�w(X), and d(X)� |X|.

We give here some examples showing the difference be-tween such cardinal functions. The Sorgenfrey line S is de-fined to be the real line R as a set. The base for S is a familyconsisting of all half open intervals of the form [p,q) withp,q ∈R and p < q . S satisfies L(S)= ω < 2ω =w(S). LetD be an uncountable discrete space and X be the one-pointcompactification of D. Then L(X)= ω < |D| = c(X). Let

Y = I exp expω (or {0,1}expexpω)be the Tychonoff product of exp expω many copies of theunit interval I = [0,1] (or the two point discrete space{0,1}), then c(Y ) = L(Y ) = ω, d(Y ) = expω and w(Y ) =exp expω.

The cardinal functions defined above are called globalcardinal functions. Now we introduce local cardinal func-tions. Let X be a topological space and x ∈ X. Recall that


a family U of neighbourhoods of x is called a neighbour-hood base of x in X if for every neighbourhood V of x ,there is U ∈ U with U ⊂ V . The character χ(x,X) ofx in X is defined by χ(x,X) = min{|U |: U is a neigh-bourhood base of x in X} + ω. Furthermore the characterχ(X) of X is defined by χ(X) = sup{χ(x,X): x ∈ X}. Inother words, χ(X) = min{κ : κ � ω, any point x ∈ X has aneighbourhood base of cardinality � κ}. A space X satis-fies the first axiom of countability, or is first-countable ifχ(X) � ω, i.e., if every point of X has a countable neigh-bourhood base. The tightness t (x,X) of x in X is definedby t (x,X)=min{κ : κ � ω, for any set A⊂X with x ∈ clA,there is a subset B of A with |B|� κ and x ∈ clB}, and de-fine the tightness t (X) of X by t (X)= sup{t (x,X): x ∈X}.In other words, t (X) = min{κ : κ � ω, for any point x ∈ Xand any subset A of X with x ∈ clA, there is a subset Bof A of cardinality � κ satisfying x ∈ clB}. Every metriz-able space satisfies the first axiom of countability, and everyfirst-countable space has countable tightness.

For compact spaces, some cardinal functions can be char-acterized by the existence of a family with some weaker con-dition. A family N of subsets of a space (X, τ) is called anetwork if every member of τ is a union of a subfamily ofN , i.e., if for any V ∈ τ and any x ∈ V , there is A ∈ Nwith x ∈ A ⊂ V . Hence, a family N of subsets of X is abase for X if and only if it is a network for X consistingof open sets. Define the network weight nw(X) of X bynw(X) = min{|N |: N is a network for X} + ω. It is inter-esting to see that w(X)= nw(X) for any compact space X.Since the family {{x}: x ∈X} is a network for any space X,we have w(X) = nw(X) � |X| for any compact space X.Note that the equality w(X) = nw(X) also holds for anymetrizable space X. Define the pseudocharacter ψ(x,X)of x in X by ψ(x,X)=min{|U |: U is a family of neighbour-hoods of x with {x} =⋂U}+ω. The pseudocharacter ofXis defined by ψ(X)= sup{ψ(x,X): x ∈X}. Then, by the de-finition of compactness, it is easy to see that χ(X)= ψ(X)for any compact T2-space X.

Some cardinal functions are bounded by some powerscontaining other cardinal functions. For example, we have|X| � expw(X) for any T1-space X. This is because eachpoint can be decided by the family of all open sets contain-ing the point.

It can be shown that w(X) � expd(X) for any regularspace X. Indeed, let B be a base and D a dense set of aregular space X. Then {int clB: B ∈ B} is a base consistingof regular open sets, where a regular open set is an openset U with U = int clU . Since each regular open set can bedecided by its intersection with D, we have the inequality.

Some global cardinal functions are bounded by somecombinations of global functions and local functions. Forexample, we have |X| � d(X)χ(X) for any T2-space X. In1969, Arkhangel’skiı̆ proved a highly nontrivial result thatthe cardinality of any compact first-countable T2-space is� 2ω, answering Alexandroff and Urysohn’s problem thathad been unanswered for about thirty years. More generally,

it can be shown that

|X|� exp(L(X) ·ψ(X) · t (X))

for any T2-space X. His original proof is difficult to under-stand. We give here the idea of a simplified proof due toR. Pol. Let X be a Lindelöf first-countable T2-space. Weshow that |X| � 2ω. Let Ux be a countable neighbourhoodbase for each x ∈X, and

A= {A: A is a subset of X with |A|� 2ω}.

Note that clA ∈ A for any A ∈ A, because each point x ∈clA is decided by a sequence in A converging to x andthere are � 2ω many sequences in A. Consider an opera-tion Φ which assigns to each closed set A ∈A, an elementΦ(A) ∈A with the following property: A⊂ Φ(A); and forany countable subfamily V of

⋃x∈AUx of X with A⊂

⋃V

and X −⋃V �= ∅, we have Φ(A)−⋃V �= ∅ (the Lindelöfproperty is used to show the existence of such V wheneverX − A �= ∅). An operation Φ exists because there are only� 2ω many choices of such V and so we can pick a point xVfrom X −⋃V for such V . Let Φ(A) be the union of A andthe set of such xV ’s. Now take any point x0 ∈ X and defineA0 = {x0}. By using transfinite induction, for each α < ω1,define Aα = cl(Φ(cl(⋃β

a-3 Cardinal functions, Part I 13

axiom of countability were studied. However uncountabilitynaturally appears in such a situation. An easy way to de-scribe this phenomenon is to consider sequential fans. Thesequential fan S(κ) with κ spines is defined by the spaceobtained from the topological sum T (κ) of κ many copies ofthe convergent sequence by identifying all the limit pointsto a point denoted by ∞. Let f :T (κ)→ S(κ) be the quo-tient map. If κ = ω, then f is a closed map from a second-countable (hence, a separable metrizable) space T (ω) onto aspace S(ω). But S(ω) is not first-countable at ∞. The char-acter χ(∞, S(ω)) is equal to the cardinal denoted by d, andit can be shown that we cannot prove d = 2ω or d < 2ω byusing our usual mathematics (i.e., ZFC set theory) (see [KV,Chapter 3]).

(2) subspaces: A cardinal function ϕ is a monotone car-dinal function if ϕ(Y ) � ϕ(X) for any subspace Y of X.Weight w and character χ are monotone. But Lindelöf num-ber L, density d and cellularity c are not monotone. For ex-ample, let D be an uncountable discrete space and X a one-point compactification of D. Then L(D) > L(X). Let Y bethe Niemytzki Plane and Z the x-axis of the plane. Then Zis a closed discrete set of Y and

d(Z)= c(Z)= 2ω > ω= d(Y )= c(Y ).

For each cardinal function ϕ which is not monotone, de-fine a new cardinal function hϕ by hϕ = sup{ϕ(Y ):Y ⊂X}.Note that hc(X) = he(X) = s(X) for any space X. hL(X)and hd(X) will be discussed in Cardinal functions Part II.

(3) products: Let ϕ be a cardinal function and κ,λ car-dinals. The property ϕ � κ is said to be a λ-multiplicativeproperty if we have ϕ(

∏α


known that compact topological groups are dyadic. For anydyadic spaceX, we have c(X)= ω and surprisingly, we havew(X)= χ(X)= t (X).

References

[1] A.V. Arkhangel’skii, Topological Function Spaces,Kluwer Academic, Dordrecht (1992).

[2] A. Dow, An introduction to applications of elementarysubmodels to topology, Topology Proc. 13 (1988), 17–72.

[3] I. Juhász, Cardinal Functions in Topology – Ten YearsLater, Mathematical Centre Tracts, Math. Centrum,Amsterdam (1983).

[4] K. Kuratowski, Topology, Vol. I, Academic Press, NewYork (1966).

[5] R.A. McCoy and I. Ntantu, Topological Propertiesof Spaces of Continuous Functions, Lecture Notes inMath., Vol. 1315, Springer, Berlin (1988).

[6] S. Todorčević, Chain-condition methods in topology,Topology Appl. 101 (2000), 45–82.

Kenichi TamanoYokohama, Japan

a-4 Cardinal functions, Part II 15

a-4 Cardinal Functions, Part II

Here in Cardinal functions II, several topics on cardinalfunctions are introduced. Some of them are rather set the-oretical, where set theoretical notions, for example, CH (theContinuum Hypothesis), MA (Martin’s Axiom), large car-dinals, the forcing method, V = L etc, play important roles.However we do not assume that readers are familiar withsuch notions. The set theoretical background can be foundin [Ku] and [7].

1. π -weight, π -character

(See [KV, Chapter 1] and [1].) The cardinal functionπ -weight is interesting not only from the topological pointof view, but also from the Boolean algebraic point of view.A family U of non-empty open sets of a space X is calleda π -base if for any non-empty open set V of X, there isU ∈ U with V ⊂U . Define the π -weight πw(X) of X by

πw(X)=min{|U |: U is a π-base for X}+ω.In terms of the theory of Boolean algebras, π -weight is “thedensity” (in the sense of Boolean algebras) of the Booleanalgebra RO(X) of all regular open sets of X. In such a way,results on topologies can be translated to results on Booleanalgebras and vice versa (see [11, 12]).

A family U of non-empty open sets of X is called a localπ -base of x in X if for every neighbourhood V of x , thereis U ∈ U with U ⊂ V . Note that elements of a local π -baseof x need not contain x .

The π -character πχ(x,X) of x in X is defined by

πχ(x,X)=min{|U |: U is a local π-base of x in X}+ω.Furthermore the π -character πχ(X) of X is defined byπχ(X)= sup{πχ(x,X): x ∈X}.

We have d(X) � πw(X) � w(X) for any topologicalspace X.

Unlike weight and character, π -weight and π -char-acter are not monotone. For example, while πw(βω) =πχ(βω) = ω, we have πw(βω − ω) � πχ(βω − ω) �cf(c) > ω, where βω is the Čech–Stone compactificationof the countable discrete space ω and cf(c) is the cofinalityof the cardinal c (see [KV, Chapter 11, 4.4.3]).

For any topological group G, we have πw(G) = w(G)and πχ(G)= χ(G) see [KV, Chapter 24, §3]).

2. Tightness

The tightness t (X) of a space X is defined as follows: Forany x ∈X, define t (x,X)=min{κ : κ � ω, for any set A⊂

X with x ∈ clA, there is a subset B of A with |B|� κ andx ∈ clB}, and define t (X) = sup{t (x,X): x ∈X}. Let α bean ordinal. A sequence of points of a space X is called a freesequence of length α in X if for each β < α (cl{xγ : γ <β}) ∩ (cl{xγ : γ � β}) = ∅. Define F(X) = sup{κ : κ is acardinal and there is a free sequence of length κ in X} +ω.

Arkhangel’skiı̆ proved that t (X)= F(X) � s(X) for anycompact space X, and Šhapirovskiı̆ proved that t (X) =hπχ(X) for any compact space X (see [KV, Chapter 1, §7]and [1]).

Now we consider the tightness of products of spaces.Tightness behaves well for compact spaces. Indeed, if{Xi : i < n} is a finite family of compact spaces with t (Xi)�κ for each i < n, then t (

∏i


rable space which is not hereditarily Lindelöf is equivalentto the existence of a regular T1 hereditarily separable spacewhich is not Lindelöf.

A space X is called right separated (respectively leftseparated) if there is a well ordering < of X such that{y ∈X: y < x} is open (respectively closed) for any x ∈X.

A linearly ordered topological space X is called aSouslin line if it has the countable chain condition (ccc),i.e., c(X)� ω, but it is not separable. It is known thatc(X) = hc(X) = hL(X) for any linearly ordered topolo-gical space. Hence every Souslin line is an L-space. It isknown that the Souslin hypothesis (= the hypothesis thatthere are no Souslin trees) is equivalent to the hypothesisthat there are no linearly ordered L-spaces. Under the ex-istence of the Souslin line, Rudin constructed an S-space.Hajnal and Juhász constructed an L-space under CH (theContinuum Hypothesis) (see [KV, Chapter 7]).

Todorčević proved that under PFA (the Proper ForcingAxiom), there are no S-spaces. In other words, every regu-lar T1 hereditarily separable space is Lindelöf. It is still un-known whether “there are no L-spaces” is consistent (i.e.,cannot be disproved by our usual axioms of ZFC) or not. Ithad been believed for a long time that the S-space problemand the L-space problem are the same problem, i.e., if thereis an S-space in some model of set theory, then there is anL-space in the same model and vice versa. But this is nottrue. Indeed, Todorčević showed that there is a model of settheory with Martin’s Axiom where there is an L-space butthere are no S-spaces (see [14]).

Martin’s Axiom and “the S and L problem” can be re-garded as a kind of partition problem (see [14], [2]). A basicform of a partition problem is the following Ramsey’s theo-rem: If there are six people, then either three of them mutu-ally know each other or three of them mutually do not knoweach other.

For any set X, define [X]2 = {{a, b}: a, b ∈ X, a �= b}.Let 6 = {0,1,2,3,4,5}. The following is a mathematicalexpression of Ramsey’s theorem above: For any partitionK0 ∪K1 ⊂ [6]2, there is a subset A of 6 with cardinality 3such that either [A]2 ⊂K0 or [A]2 ⊂K1. It is interesting tosee that such a way of thinking is essential to our set theoryand topology (for the theory of partition relations, see [5], orAppendix 4 in [8]).

4. Reflection theorems

(See [6] and [3].) Generally speaking, a reflection theoremis a theorem of the form “if a set (or a space) X has a prop-erty P , then there is a subset (or a subspace) Y of a small size(in some sense) satisfying P ”. Reflection principle plays animportant key role in set theory.

In topology, Hajnal–Juhász proved the following beauti-ful reflection theorem: If X does not have a countable base,then there is a subspace of X of cardinality � ω1 which doesnot have a countable base. To state a more general result dueto them, we need a definition. Let ϕ be a cardinal function

and C a class of spaces. Then ϕ is said to reflect a cardi-nal κ if whenever ϕ(X) � κ , we have Y ⊂ X with |Y |� κand ϕ(Y )� κ . For a class C of spaces, ϕ reflects a cardinal κfor C if whenever X ∈ C and ϕ(X)� κ , we have Y ⊂X with|Y |� κ and ϕ(Y ) � κ . By using this definition, we can re-state Hajnal–Juhász’s result as follows: weightw reflects ω1.Furthermore they showed that weight w reflects any infinitecardinal.

Cardinal functions cellularity c, extent e, spread s, hered-itary density hd and hereditary Lindelöf degree hL reflectsall infinite cardinals. Density d reflects every regular car-dinal but need not reflect a singular cardinal for the class ofT1-spaces. For the class of compact Hausdorff spaces, tight-ness t and character χ reflects any infinite cardinal.

The main technique to show reflection theorems is aclosing-off argument (in other words, a method by using ele-mentary submodels) mentioned in Cardinal functions, Part I.

5. sup=max problem

Recall that the cellularity c(X) is defined by c(X) =min{κ : κ � ω, every family of pairwise disjoint nonemptyopen sets of X has cardinality � κ}. In other words, c(X)=sup{κ : there is a pairwise disjoint family of cardinality κconsisting of open sets of X} +ω. A natural question arises:Can the “supremum” in this definition be replaced by the“maximum”? In other words, if c(X)= κ , then does a pair-wise disjoint family of open sets of cardinality κ really exist?Such kind of problems are called “sup = max problems”.Note that such a question is not trivial only when κ is a limitcardinal (i.e., a singular cardinal or a regular limit cardinal).Erdös–Tarski proved that if c(X) = κ for a singular cardi-nal κ , then X has a pairwise disjoint family of cardinalityκ consisting of nonempty open sets. For other results onsup = max problems, see [KV, Chapter 1, §12], [9, Chap-ter 4].

6. The number of open sets, compact sets, etc

Let o(X) be the number of open sets of X, C(X) the fam-ily of all continuous real valued functions, and RO(X)the family of all regular open sets of X. Here a subsetU of X is called a regular open set if U = int clU . It isknown that |C(X)|ω = |C(X)|, and |RO(X)|ω = |RO(X)|for any infinite Hausdorff space (see [KV, Chapter 1, §10]).It is natural to ask whether o(X)ω = o(X) or not. For ametrizable space X, it can be shown that o(X) = 2w(X),hence this equality holds. In 1986, Shelah proved that ifX is an infinite compact T2-space, then o(X)ω = o(X)(see [10]).

Let K(X) be the family of all compact subsets of aspace X. For a T2-space X, we have |K(X)| � 2hL(X) and|K(X)|� 2e(X)·ψ(X) (see [KV, Chapter 1, §9]).

a-4 Cardinal functions, Part II 17

References

[1] A.V. Arkhangel’skiı̆, Structure and classification oftopological spaces and cardinal invariants, RussianMath. Surveys 33 (1978), 33–96.

[2] M. Bekkali, Topics in Set Theory, Lecture Notes inMathematics, Springer, Berlin (1991).

[3] A. Dow, An introduction to applications of elementarysubmodels to topology, Topology Proc. 13 (1988), 17–72.

[4] K. Eda, G. Gruenhage, P. Koszmider, K. Tamano andS. Todorčević, Sequential fans in topology, TopologyAppl. 67 (1995), 189–220.

[5] P. Erdös, A. Hajnal, A. Máté and R. Rado, Combina-torial Set Theory: Partition Relations for Cardinals,North-Holland, Amsterdam (1984).

[6] R.E. Hodel and J.E. Vaughan, Reflection theorems forcardinal functions, Topology Appl. 100 (2000), 47–66.

[7] T. Jech, Set Theory, Springer, Berlin (1997).[8] I. Juhász, Cardinal Functions in Topology, Mathemati-

cal Centre Tracts, Math. Centrum, Amsterdam (1971).

[9] I. Juhász, Cardinal Functions in Topology—Ten YearsLater, Mathematical Centre Tracts, Math. Centrum,Amsterdam (1983).

[10] I. Juhász, Cardinal functions, Recent Progress in Gen-eral Topology, M. Hušek and J. van Mill, eds, North-Holland, Amsterdam (1992), 417–441.

[11] S. Koppelberg, Handbook of Boolean Algebras, Vol. 1,J.D. Monk, ed., North-Holland, Amsterdam (1989).

[12] J.D. Monk, Cardinal Invariants on Boolean Algebras,Birkhäuser, Basel (1996).

[13] M.E. Rudin, S and L spaces, Surveys in General To-pology, G.M. Reed, ed., Academic Press, New York(1980), 431–444.

[14] S. Todorčević, Partition Problems in Topology, Amer.Math. Soc., Providence, RI (1989).

Kenichi TamanoYokohama, Japan


a-5 Convergence

A sequence in X is a map from the set N of natural numbersto X. A sequence s :N→X will often be identified with itsimage {s(n): n ∈N}, and we will use a shorter notation (sn)to refer to sequences. A sequence (tn) is a subsequence ofa sequence (sn) if there exists an increasing map k :N→ Nsuch that tn = sk(n) for all n. A sequence (xn) of points ofa topological space is a convergent sequence with limit x ,if for every neighbourhood W of x there exists a naturalnumber nW such that xn ∈W for each n� nW . Convergenceof sequences in a topological space X satisfies the followingthree conditions:

(S1) if a sequence (xn) converges to x then every subse-quence of (xn) also converges to x ,

(S2) if xn = x for each n then (xn) converges to x ,(S3) if a sequence (xn) does not converge to x then there

exists a subsequence (yn) of (xn) such that no subse-quence of (yn) converges to x .

Convergence of sequences can be introduced axiomati-cally. In [7] M. Fréchet defined an L-space as a set X and arelation “(xn) converges to x” (in symbols, x = limn xn) be-tween sequences (xn) of elements of X and elements x of Xsatisfying conditions (S1), (S2) and the following condition

(S0) a sequence (xn) cannot converge to two differentpoints.

An L∗-space is an L-space that fulfills (S3), the additionalaxiom of P. Alexandroff and P. Urysohn [2]. An S∗-space isan L∗-space satisfying the following diagonal condition:(S4) x = limn xn and xn = limk xn,k for every n implies

that there exist an increasing sequence (nm) of naturalnumbers and a sequence (km) of natural numbers withx = limm xnm,km .

A subset S of an L-space is closed whenever the limit ofevery convergent sequence of points of S belongs to S; a setis open if it is the complement of a closed set. The collectionof open subsets of an L-space is a (not necessarily Haus-dorff) topology. Conversely, the convergent sequences in aHausdorff topological space X constitute an L∗-space gen-erated by X. A subset S of a topological space X is sequen-tially closed (respectively, sequentially open) if it is closed(respectively, open) with respect to the L∗-space generatedby X. A topological space is called sequential if every se-quentially open set is open. A topological space is a Fréchetspace or a Fréchet–Urysohn space if x ∈ clA (where clAdenotes the closure of a subset A) implies the existence ofa sequence of points from A that converges to x . The con-vergent sequences of a Hausdorff Fréchet topology consti-tute an S∗-space; conversely, each S∗-space determines aFréchet (not necessarily a Hausdorff) topology.

The concept of net, introduced by E.H. Moore andH.L. Smith in [15], is an extension of that of sequence.A partially ordered set (D,�) is said to be directed if forevery α,β ∈D there exits δ ∈D such that α � δ and β � δ.If (D,�) is a directed set, then a map t :D→ X is calleda net on X. Every sequence can be identified with a net de-fined on (N,�), the set of natural numbers directed by thenatural order. A net is constant if the corresponding map isconstant. A net s :E→X is a subnet of t :D→X if thereexists a map f :E→ D such that s = t ◦ f and for everyδ ∈ D there is ε ∈ E with f (ε) � δ. If X is a topologicalspace, then a net t :D→X converges to x ∈X (in symbols,x ∈ lim t = limδ t (δ)) provided that for every neighbourhoodW of x there is δW ∈D such that t (α) ∈W for each α � δW .A sequence (xn) in a topological space converges to x if andonly if the net t :N → X defined by t (n) = xn convergesto x .

Let (D,�D) and (Eδ,�δ) for each δ ∈D be directed sets.For (δ,ϕ), (δ′, ϕ′) ∈D×∏δ∈D Eδ define (d,ϕ)� (d ′, ϕ′) ifand only if d �D d ′ and ϕ(δ)�δ ϕ′(δ) for each δ ∈D. Then(D ×∏δ∈D Eδ,�) is a directed set. If fδ :Eδ →X is a netfor each δ ∈D, then the map

�{fδ : δ ∈D} :D ×∏δ∈D

Eδ →X

defined by�{fδ : δ ∈D}(d,ϕ)= fd(ϕ(d)) is a net called thediagonal net of the family {fδ : δ ∈D}.

Convergence of nets in a topological space is often calledMoore–Smith convergence and has the following proper-ties (N1)–(N4) similar to properties (S1)–(S4), respectively:

(N1) if a net converges to x , then every subnet converges tox ,

(N2) every constant net converges to its constant value,(N3) if x /∈ lim t then there exists a subnet s of t such that

x /∈ lim r for each subnet r of s, and(N4) if x ∈ lim t and t (δ) ∈ limfδ for each δ ∈D, then x ∈

lim�{fδ : δ ∈D}.A non-empty family F of subsets of a set X is called a

filter on X if: (i) F ∈ F and F ⊂ G ⊂ X implies G ∈ F ,(ii) F0 ∩ F1 ∈ F provided that F0,F1 ∈ F , and (iii) ∅ /∈ F .The family N (x) of neighbourhoods of x is a filter for eachpoint x of a topological space: its neighbourhood filter.A family C of non-empty sets is a centered family pro-vided that C1,C2 ∈ C implies the existence of C3 ∈ C withC3 ⊂ C1 ∩C2. If C is a centered family of subsets of X, thenF = {F ⊆ X: ∃C ∈ C, C ⊆ F } is a filter on X generatedby C (and we say that C generates F ). A centered subfamilyB of a filter F is a filter base provided that B generates F .

a-5 Convergence 19

For every sequence (xn) of elements of a set X, the family{{xk: k � n}: n ∈N} is centered, and thus generates a filter onX called the filter generated by the sequence (xn). Maxi-mal filters (with respect to the set inclusion) are called ultra-filters. For a subset A⊂ X, the filter Ȧ= {B ⊂ X: A⊂ B}is called the principal filter of A. A free filter is a filter withempty intersection.

If X is a set, then we use F(X) to denote the set of allfilters on X. If Φ :Y → F(X) is a map and F ∈ F(Y ), then

Φ(F )=⋃F∈F

⋂y∈F

Φ(y) ∈ F(X)

is the sum or contour of Φ over F ; in other words, A ∈Φ(F ) whenever there exists F ∈F such that A ∈Φ(y) forevery y ∈ F .

A subset ξ of F(X)×X is called a filter convergence (or aconvergence structure) on a set X provided that (F , x) ∈ ξand F ⊂ G imply (G, x) ∈ ξ and (ẋ, x) ∈ ξ for every x ∈X,where ẋ stands for the principal filter of {x}. If (F , x) ∈ ξ ,then we say that F converges to x (x is a limit of F ) with re-spect to ξ . We often write x ∈ limξ F instead of (F , x) ∈ ξ ,and when ξ has been fixed, simply x ∈ limF . In other words,lim represents a filter convergence on X if it satisfies

(F1) if F ,G ∈ F(X), F ⊂ G and x ∈ limF , then x ∈ limG,and

(F2) x ∈ lim ẋ for every x ∈X.(The original definition of convergence structure in [13] byH.-J. Kowalsky included also the condition lim(F0 ∩F1)⊃limF0 ∩ limF1 for each filters F0 and F1.) By a conver-gence space we understand a set and a filter convergence onthat set. A filter convergence is a Hausdorff convergence iflimF is at most a singleton for every filter F . We say F is aconvergent filter if limF �= ∅. The set limF is often calledthe limit set of F .

Every L-space determines a filter convergence as follows:x ∈ limF if and only if there exists a sequence (xn) that con-verges to x such that F contains the filter generated by (xn).Filter convergences determined by L-spaces are Hausdorff(due to the condition (S0)).

Let X be a topological space. A filter F on X convergesto a point x ∈X if every neighbourhood of x belongs to F .This special filter convergence satisfies (F1), (F2) and also

(F3)⋂{limF : F ∈ F} ⊂ lim⋂{F : F ∈ F} for every F ⊂F(X), and

(F4) limF ⊂ limΦ(F ) for every F ∈ F(X) and each mapΦ :X→ F(X) such that x ∈ limΦ(x) for every x ∈X.

For a topological space X, convergence of nets and filterconvergence are equivalent. Indeed, if (D,�) is a directedset and s :D → X is a net, then the family {Bδ : δ ∈ D},where Bδ = {s(α): α ∈D, α � δ}, is centered, and thus gen-erates a filter F on X. The net s converges to x ∈ X if andonly if the filter F converges to x . Conversely, if F is a filteron X, then the set

D = {(x,F ): x ∈ F}⊂X×F

partially ordered by (x0,F0) � (x1,F1) whenever F0 ⊃ F1,is a directed set. The map s :D→X defined by s(x,F )= xis a net that converges to some element x ′ ∈ X if and onlyif the filter F converges to x ′. If the filter corresponding toa net is an ultrafilter then the net itself is often called anultranet or a universal net.

If T is a topology onX, thenCT denotes the filter conver-gence for which x ∈ limCT F if and only if O ∈F for everyO ∈ T such that x ∈ O . Conversely, every filter conver-gence ξ defines a topology Oξ by declaring O ∈Oξ when-ever limξ F ∩ O �= ∅ implies that O ∈ F . A filter conver-gence ξ is said to be topological (or simply, a topology) ifξ = COξ . A convergence is topological if and only if it ful-fills (F1) through (F4). A filter convergence is: (i) a pseudo-topology if x ∈ limF provided that x ∈ limU for every ul-trafilter U ⊃F , (ii) a paratopology if x /∈ limF implies theexistence of a filter H with a countable filter base such thatF ∩H �= ∅ whenever F ∈ F and H ∈H, and x /∈ limG forevery G ⊃H, (iii) a pretopology, or a Moore closure space,if (F3) holds, and (iv) diagonal if it fulfills (F4). Hence afilter convergence is a topology if and only if it is a diagonalpretopology. The notion of pretopology is due to F. Haus-dorff [11], while that of pseudotopology to G. Choquet [3].

Properties of convergent filters correspond to some prop-erties of convergent nets, namely (N1) amounts to (F1),while (N2) is equivalent to (F2). The diagonal condition(N4) is equivalent to the conjunction of (F3) and (F4). Fi-nally, (N3) is equivalent to being a pseudotopology and isimplied by (N4).

Given a filter convergence ξ on a set X, a subset A of X isξ -compact if for every filter H such thatA ∈H there exists afilter G ⊃F such that limξ G∩A �= ∅. IfX is ξ -compact thenwe call ξ a compact convergence and if every ξ -convergentfilter contains a ξ -compact set then ξ is a locally compactconvergence. A convergence is sequential (respectively, ofcountable character) if x ∈ limF implies the existence ofa filter E generated by a sequence (respectively, having acountable filter base) such that E ⊂F and x ∈ limE .

If f :X→ Y and F is a filter on X, then f (F ) denotesthe filter on Y generated by {f (F ): F ∈F}. If X and Y areconvergence spaces, then a map f :X→ Y is continuousif f (limF ) ⊂ limf (F ) for every filter F on X. A filterconvergence ξ is finer than a filter convergence τ , and τ iscoarser than ξ (in symbols, ξ � τ ), if the identity map fromξ to τ

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