DOCSLIB.ORG

4Rends Ferences Aim Without 0Roposals "Irkhëuser 0

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
4Rends Ferences Aim Without 0Roposals 42%.$3).-!4(%-!4)#3 4RENDSIN-ATHEMATICSISASERIESDEVOTEDTOTHEPUBLICATIONOFVOLUMESARISINGFROMCON FERENCESANDLECTURESERIESFOCUSINGONAPARTICULARTOPICFROMANYAREAOFMATHEMATICS)TS AIMISTOMAKECURRENTDEVELOPMENTSAVAILABLETOTHECOMMUNITYASRAPIDLYASPOSSIBLE WITHOUTCOMPROMISETOQUALITYANDTOARCHIVETHESEFORREFERENCE 0ROPOSALSFORVOLUMESCANBESENTTOTHE-ATHEMATICS%DITORATEITHER "IRKHËUSER6ERLAG 0/"OX #( "ASEL 3WITZERLAND OR "IRKHËUSER"OSTON)NC -ASSACHUSETTS!VENUE #AMBRIDGE -! 53! -ATERIALSUBMITTEDFORPUBLICATIONMUSTBESCREENEDANDPREPAREDASFOLLOWS !LLCONTRIBUTIONSSHOULDUNDERGOAREVIEWINGPROCESSSIMILARTOTHATCARRIEDOUTBYJOURNALS ANDBECHECKEDFORCORRECTUSEOFLANGUAGEWHICH ASARULE IS%NGLISH!RTICLESWITHOUT SURVEYPAPERS HOWEVER AREWELCOME 7EEXPECTTHEORGANIZERSTODELIVERMANUSCRIPTSINAFORMTHATISESSENTIALLYREADYFORDIRECT REPRODUCTION!NYVERSIONOF4% ONEPARTICULARDIALECTOF4% HËUSER &URTHERMORE INORDERTOGUARANTEETHETIMELYAPPEARANCEOFTHEPROCEEDINGSITISESSENTIAL EACHARTICLEWILLRECEIVEFREEOFFPRINTS4OTHEPARTICIPANTSOFTHECONGRESSTHEBOOK WILLBEOFFEREDATASPECIALRATE 4OPICSIN#OHOMOLOGICAL3TUDIE3ET4HEORY S OF!LGEBRAIC6ARIETIE#ENTREDE2ECERCA-ATEMÌTICA S "ARCELONA n )MPANGA,ECTURE.OTES 0I-RDQ%DJDULDOTR0RAGACZ %6WHYR7RGRUFHYLFDITOR (GLWRUV "IRKHËUSER6ERLAG "ASELs "OSTON s "ERLIN %D%DITORSITOR 0IOTR0RAGACZ*OAN"AGARIA 3TEVO4ODORCEVIC )NSTITUTEOF-ATHEMATICSOFTHE$EPARTAMENTDE,ÛGICA $EPARTMENTOF-ATHEMATICS 0OLISH!CADEMYOF3CIENCES(ISTÛRIAI&ILOSOlADELA#IÒNCIA 5NIVERSITYOF4ORONTO UL3NIADECKICH5NIVERSITATDE"ARCELONA 4ORONTO -3% #ANADA 0/"OX"ALDIRI2EIXACSN E MAILSTEVO MATHTORONTOEDU 7ARSZAWA"ARCELONA 3PAIN 0OLANDE [email protected] AND E MAILPPRAGACZ IMPANGOVPL 5NIVERSITÏ0ARIS #.23 5-2 0LACE*USSIEU #ASE 0ARIS#EDEX &RANCE - . ! !XX $XX %XX 1 3XX 0 + ( !-ATHEMATICAL3UBJECT#LASSIlCATION%XX /% % #)0CATALOGUERECORDFORTHISBOOKISAVAILABLEFROMTHE % % % ,% %IBRARYOF#ONGRESS 7ASHINGTON$# 53! "!#)0CATALOGUERECORDFORTHISBOOKISAVAILABLEFROMTHEIBLIOGRAPHICINFORMATIONPUBLISHEDBY$IE$EUTSCHE"IBLIOTHEK ,IBRARYOF#ONGRESS 7ASHINGTON$# 53! EDETAILED BIBLIOGRAPHICDATAISAVAILABLEINTHE)NTERNETATHTTPDNBDDBDE "IBLIOGRAPHICINFORMATIONPUBLISHEDBY$IE$EUTSCHE"IBLIOTHEK $IE$EUTSCHE"IBLIOTHEKLISTSTHISPUBLICATIONINTHE$EUTSCHE.ATIONALBIBLIOGRAlEDETAILED BIBLIOGRAPHICDATAISAVAILABLEINTHE)NTERNETATHTTPDNBDDBDE )3". "IRKHËUSER6ERLA G "ASELn"OSTONn"ERLIN 4HISWORKISSUBJECTTOCOPYRIGHT!LLRIGHTSARERESERVED WHETHERTHEWHOLEORPARTOFTHE BANKS&ORANYKINDOFUSEPERMISSIONOFTHECOPYRIGHTOWNERMUSTBEOBTAINED ¥"IRKHËUSER6ERLA G 0/"OX #( "ASEL 3WITZERLAND 0ARTOF3PRINGER3CIENCE "USINESS-EDIA 0RINTEDONACID FREEPAPERPRODUCEDFROMCHLORINE FREEPULP4#& df 0RINTEDIN'ERMANY )3". E )3". )3". WWWBIRKHAUSERCH Contents Foreword ................................................................. vii Survey Papers J. Bagaria, N. Castells and P. Larson AnΩ-logicPrimer ................................................... 1 M. Bekkali and D. Zhani UpperSemi-latticeAlgebrasandCombinatorics ...................... 29 R. Bosch SmallDefinably-largeCardinals ...................................... 55 A. E. Caicedo Real-valued Measurable Cardinals and Well-orderings oftheReals ......................................................... 83 A. Marcone Complexity of Sets and Binary Relations in Continuum Theory: A Survey ............................................................ 121 A.R.D. Mathias WeakSystemsofGandy,JensenandDevlin ......................... 149 B. Tsaban Some New Directions in Infinite-combinatorial Topology . 225 Research Papers T. Banakh and A. Blass The Number of Near-Coherence Classes of Ultrafilters is Either Finite or 2c ................................................... 257 S.-D. Friedman StableAxiomsofSetTheory ........................................ 275 S.-D. Friedman ForcingwithFiniteConditions ....................................... 285 G. Hjorth Subgroups of Abelian Polish Groups . 297 vi Contents P. Koepke and P. Welch OntheStrengthofMutualStationarity .............................. 309 P. Matet Part(κ, λ)andPart∗(κ, λ) ........................................... 321 C. Morgan LocalConnectednessandDistanceFunctions ........................ 345 R. Schindler Bounded Martin’s Maximum and Strong Cardinals . 401 Foreword This is a collection of articles on set theory written by some of the participants in the Research Programme on Set Theory and its Applications that took place at the Centre de Recerca Matem`atica (CRM) in Bellaterra (Barcelona). The Programme run from September 2003 to July 2004 and included an international conference on set theory in September 2003, an advanced course on Ramsey methods in analysis∗ in January 2004, and a joint CRM-ICREA workshop on the foundations of set theory in June 2004, the latter held in Barcelona. A total of 33 short and long term visitors from 15 countries participated in the Programme. This volume consists of two parts, the first containing survey papers on some of the mainstream areas of set theory, and the second containing original research papers. All of them are authored by visitors who took part in the set theory Programme or by participants in the Programme’s activities. The survey papers cover topics as Omega-logic, applications of set theory to lattice theory and Boolean algebras, real-valued measurable cardinals, complexity of sets and relations in continuum theory, weak subsystems of axiomatic set the- ory, definable versions of large cardinals, and selection theory for open covers of topological spaces. As for the research papers, they range from topics such as the number of near-coherence classes of ultrafilters, the consistency strength of bounded forcing axioms, Pκ(λ) combinatorics, some applications of morasses, subgroups of Abelian Polish groups, adding club subsets of ω2 with finite conditions, the consistency strength of mutual stationarity, and new axioms of set theory. We would like to thank all participants in the Programme and its related activities for their effort in making these very successful ventures. We also want to thank the CRM Director and its staff, as well as all the funding institutions, for making the Programme possible. Joan Bagaria Stevo Todorcevic Editors ∗The volume: Spiros A. Argyros and Stevo Todorcevic, Ramsey Methods in Analysis. Advanced Courses in Mathematics CRM Barcelona. Birkh¨auser 2005, contains the notes of the course in an expanded form. Set Theory Trends in Mathematics, 1–28 c 2006 Birkh¨auser Verlag Basel/Switzerland An Ω-logic Primer Joan Bagaria, Neus Castells and Paul Larson Abstract. In [12], Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about Ω-logic, relative to the published liter- ature, leading up to the generic invariance of Ω-logic and the Ω-conjecture. Keywords. Ω-logic, Woodin cardinals, A-closed sets, universally Baire sets, Ω-conjecture. Introduction One family of results in modern set theory, called absoluteness results,showsthat the existence of certain large cardinals implies that the truth values of certain sen- tences cannot be changed by forcing1. Another family of results shows that large cardinals imply that certain definable sets of reals satisfy certain regularity prop- erties, which in turn implies the existence of models satisfying other large cardinal properties. Results of the first type suggest a logic in which statements are said to be valid if they hold in every forcing extension. With some technical modifica- tions, this is Woodin’s Ω-logic, which first appeared in [12]. Results of the second type suggest that there should be a sort of internal characterization of validity in Ω-logic. Woodin has proposed such a characterization, and the conjecture that it succeeds is called the Ω-conjecture. Several expository papers on Ω-logic and the Ω-conjecture have been published [1, 13, 14, 15, 16, 17]. Here we briefly discuss The first author was partially supported by the research projects BFM2002-03236 of the Minis- terio de Ciencia y Tecnolog´ıa, and 2002SGR 00126 of the Generalitat de Catalunya.Thethird author was partially supported by NSF Grant DMS-0401603. This paper was written during the third author’s stay at the Centre de Recerca Matem`atica (CRM), whose support under a Mobil- ity Fellowship of the Ministerio de Educaci´on, Cultura y Deportes is gratefully acknowledged. It was finally completed during the first and third authors’ stay at the Institute for Mathematical Sciences, National University of Singapore, in July 2005. 1Throughout this paper, by “forcing” we mean “set forcing”. 2 J. Bagaria, N. Castells and P. Larson the technical background of Ω-logic, and prove some of the basic theorems in this area. This paper assumes a basic knowledge of Set Theory, including constructibil- ity and forcing. All undefined notions can be found in [4]. 1. Ω 1.1. Preliminaries Given a complete Boolean algebra B in V , we can define the Boolean-valued model V B by recursion on the class of ordinals On: V B = ∅ 0 B B Vλ = Vβ ,ifλ is a limit ordinal β<λ B { → B | ⊆ B} Vα+1 = f : X X Vα , B B B B Then, V = α∈On Vα . The elements of V are called -names. Every element x of V has a standard B-name xˇ, defined inductively by: ∅ˇ = ∅,andˇx : {yˇ : y ∈ x}→{1B}. ∈ B { ∈ | ∈ B } B For each x V ,letρ(x)=min α On x Vα+1 ,therank of x in V . Given ϕ, a formula of the language of set theory with parameters in V B,we say that ϕ is true in V B if its Boolean-value is 1B, i.e., V B ϕ iff [[ϕ]] B =1B, where [[·]] B is defined by induction
Load more

Recommended publications
  • 1.1.5 Hausdorff Spaces
    1. Preliminaries Proof. Let xn n N be a sequence of points in X convergent to a point x X { } 2 2 and let U (f(x)) in Y . It is clear from Definition 1.1.32 and Definition 1.1.5 1 2F that f − (U) (x). Since xn n N converges to x,thereexistsN N s.t. 1 2F { } 2 2 xn f − (U) for all n N.Thenf(xn) U for all n N. Hence, f(xn) n N 2 ≥ 2 ≥ { } 2 converges to f(x). 1.1.5 Hausdor↵spaces Definition 1.1.40. A topological space X is said to be Hausdor↵ (or sepa- rated) if any two distinct points of X have neighbourhoods without common points; or equivalently if: (T2) two distinct points always lie in disjoint open sets. In literature, the Hausdor↵space are often called T2-spaces and the axiom (T2) is said to be the separation axiom. Proposition 1.1.41. In a Hausdor↵space the intersection of all closed neigh- bourhoods of a point contains the point alone. Hence, the singletons are closed. Proof. Let us fix a point x X,whereX is a Hausdor↵space. Denote 2 by C the intersection of all closed neighbourhoods of x. Suppose that there exists y C with y = x. By definition of Hausdor↵space, there exist a 2 6 neighbourhood U(x) of x and a neighbourhood V (y) of y s.t. U(x) V (y)= . \ ; Therefore, y/U(x) because otherwise any neighbourhood of y (in particular 2 V (y)) should have non-empty intersection with U(x).
    [Show full text]
  • Chapter 7 Separation Properties
    Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and .
    [Show full text]
  • 9 | Separation Axioms
    9 | Separation Axioms Separation axioms are a family of topological invariants that give us new ways of distinguishing between various spaces. The idea is to look how open sets in a space can be used to create “buffer zones” separating pairs of points and closed sets. Separations axioms are denoted by T1, T2, etc., where T comes from the German word Trennungsaxiom, which just means “separation axiom”. Separation axioms can be also seen as a tool for identifying how close a topological space is to being metrizable: spaces that satisfy an axiom Ti can be considered as being closer to metrizable spaces than spaces that do not satisfy Ti. 9.1 Definition. A topological space X satisfies the axiom T1 if for every points x; y ∈ X such that x =6 y there exist open sets U;V ⊆ X such that x ∈ U, y 6∈ U and y ∈ V , x 6∈ V . X U V x y 9.2 Example. If X is a space with the antidiscrete topology and X consists of more than one point then X does not satisfy T1. 9.3 Proposition. Let X be a topological space. The following conditions are equivalent: 1) X satisfies T1. 2) For every point x ∈ X the set {x} ⊆ X is closed. Proof. Exercise. 9.4 Definition. A topological space X satisfies the axiom T2 if for any points x; y ∈ X such that x =6 y 60 9. Separation Axioms 61 there exist open sets U;V ⊆ X such that x ∈ U, y ∈ V , and U ∩ V = ?. X U V x y A space that satisfies the axiom T2 is called a Hausdorff space.
    [Show full text]
  • Theory of G-T G-Separation Axioms 1. Introduction Whether It Concerns The
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 July 2018 doi:10.20944/preprints201807.0095.v1 Theory of g-Tg-Separation Axioms Khodabocus m. i. and Sookia n. u. h. Abstract. Several specific types of ordinary and generalized separation ax- ioms of a generalized topological space have been defined and investigated for various purposes from time to time in the literature of topological spaces. Our recent research in the field of a new class of generalized separation axioms of a generalized topological space is reported herein as a starting point for more generalized classes. Key words and phrases. Generalized topological space, generalized sepa- ration axioms, generalized sets, generalized operations 1. Introduction Whether it concerns the theory of T -spaces or Tg-spaces, the idea of adding a 1 T T Tα or a g-Tα-axiom (with α =( 0, 1, 2,):::) to the axioms for a -space( T = (Ω); ) to obtain a T (α)-space T(α) = Ω; T (α) or a g-T (α)-space g-T(α) = Ω; g-T (α) or, the idea of adding a T or a -T -axiom (with α = 0, 1, 2, :::) to the axioms g,α g g,α ( ) T T T (α) (α) T (α) T (α) for a g-space T( g = (Ω; g)) to obtain a g -space Tg = Ω; g or a g- g - (α) T (α) space g-Tg = Ω; g- g has never played little role in Generalized Topology and Abstract Analysis [17, 19, 25]. Because the defining attributes of a T -space in terms of a collection of T -open or g-T -open sets or a Tg-space in terms of a collection of Tg-open or g-Tg-open sets, respectively, does little to guarantee that the points in the T -space T or the Tg-space Tg are somehow distinct or far apart.
    [Show full text]
  • A SEPARATION AXIOM BETWEEN SEMI-To and SEMI-T1 *
    Pro Mathematica: Vol. XI, Nos. 21-22, 1997 A SEPARATION AXIOM BETWEEN SEMI-To AND SEMI-T1 * Miguel Caldas Dedicated to Professor Chaim S. Honig on the occasion of his 70th birthday Abstract The author introduces a new separation axiom and studies so me of their basic properties. The implication of these new separation axiom among themselves and with the well known axioms semi-T 2 semi-T 1 and semi-T 0 are obtained. l. Introduction1 Semi-open sets where introduced and investigated by N. Levine [6] in 1963. In 1975, S.N. Maheshwari and R. Prasad [7] used semi-open sets to define and investigate three new separation axioms, called semi-T2o semi-T1 and semi-T0 . Moreover, they have shown that the following implications hold. ©> Universidade Federal Fluminense IMUFF, RJ-Brasil. AMS (1980) subject classification (1985-Revision): 54 DIO 1 Key Words: Topology, semi-open sets. map semi-contimwus map irresolute, sg-closed sets. Tz ~ semi- Tz J, J, T, ~ semi- T1 J, J, To ~ semi- T0 Later, in 1982 P.Bhattacharyya and B.K. Lahiri [1] used semi-open sets to define the axiom semi-T112 and further investigated the separation axioms semi-T2, semi-T1 and semi-T0 • For other properties of semi-T112 , see [4]. The purpose of this paper is to introduce a new separation axiom semi-D1 which is strictly between semi-T0 and semi-T" and discuss its relations with the axioms mentioned above. Listed below are definitions that will be utilized. We identify the separation axioms with the class of topological spaces satisfying these axioms.
    [Show full text]
  • Arxiv:Math/9810074V1 [Math.GN] 12 Oct 1998
    Unification approach to the separation axioms between ∗ T0 and completely Hausdorff Francisco G. Arenas,† Julian Dontchev‡and Maria Luz Puertas§ September 19, 2021 Abstract The aim of this paper is to introduce a new weak separation axiom that generalizes the separation properties between T1 and completely Hausdorff. We call a topological space (X,τ) a Tκ,ξ-space if every compact subset of X with cardinality ≤ κ is ξ-closed, where ξ is a general closure operator. We concentrate our attention mostly on two new concepts: kd-spaces and T 1 -spaces. 3 1 Introduction The definitions of most (if not all) weak separation axioms are deceptively simple. However, the structure and the properties of those spaces are not always that easy to comprehend. In this paper we try to unify the separation axioms between T0 and completely Hausdorff by introducing the concept of Tκ,ξ-spaces. We call a topological space (X, τ) a Tκ,ξ-space arXiv:math/9810074v1 [math.GN] 12 Oct 1998 if every compact subset of X with cardinality ≤ κ is ξ-closed where ξ is a given closure operator. With different settings on κ and ξ we derive most of the well-known separation properties ‘in the semi-closed interval [T0, T3)’. We are going to consider not only Kuratowski closure operators but more general closure operators, such as the λ-closure operator [1] for ∗1991 Math. Subject Classification — Primary: 54A05, 54D10; Secondary: 54D30, 54H05. Key words and phrases — θ-closed, δ-closed, Urysohn, zero-open, λ-closed, weakly Hausdorff, T0, T1, com- pletely Hausdorff, kc-space, kd-space, Tκ,λ-space, anti-compact.
    [Show full text]
  • The Separation Axioms We Give Two Examples of Spaces That Satisfy A
    F.l The separation axioms We give two examples of spaces that satisfy a given separation axiom but not the next stronger one. Te first is a familiar space, and the second is not. Teorem F.1. If J is uncountable, J is completely regular, but not normal. Proof. The proof follows the outline given in Exercise 9 of §32. The space RJ is of course completely regular, being a product of completel regular spaces. Le't X = Z+ ; since X is a closed subspace of RJ it suffices to show that X i not normal. WE! shall use functional notation for the elements of X rather than tuple notation. Given a finite subset B of J and given a point x of X, let U(x,B) be the set of all those elements y of X such that y(od = x( ) for all a in B. Then U(x,B) is open in X; irldeed, it is the cartesian product FU~, where U is a one-point set fr in B and U = 2+ otherwise. It is immediate that the sets U(x,B) form a basis for X, since the one-point sets form a basis for 7. Given a positive integer n, let Pn be the subset of X consisting of those maps x :J-47 such that for each i dfferent from n, -1 the set x (i) consists, of at most one element of J. (This of course implies that x In) consists of uncountably many elements of J.) The set Pn is closed, for if y is not in Pn, then there is an integer i n and distinct indices , " of J such that y(4) = y( ) = i.
    [Show full text]
  • 9. Stronger Separation Axioms
    9. Stronger separation axioms 1 Motivation While studying sequence convergence, we isolated three properties of topological spaces that are called separation axioms or T -axioms. These were called T0 (or Kolmogorov), T1 (or Fre´chet), and T2 (or Hausdorff). To remind you, here are their definitions: Definition 1.1. A topological space (X; T ) is said to be T0 (or much less commonly said to be a Kolmogorov space), if for any pair of distinct points x; y 2 X there is an open set U that contains one of them and not the other. Recall that this property is not very useful. Every space we study in any depth, with the exception of indiscrete spaces, is T0. Definition 1.2. A topological space (X; T ) is said to be T1 if for any pair of distinct points x; y 2 X, there exist open sets U and V such that U contains x but not y, and V contains y but not x. Recall that an equivalent definition of a T1 space is one in which all singletons are closed. In a space with this property, constant sequences converge only to their constant values (which need not be true in a space that is not T1|this property is actually equivalent to being T1). Definition 1.3. A topological space (X; T ) is said to be T2, or more commonly said to be a Hausdorff space, if for every pair of distinct points x; y 2 X, there exist disjoint open sets U and V such that x 2 U and y 2 V .
    [Show full text]
  • An Overview of Separation Axioms in Recent Research
    International Journal of Pure and Applied Mathematics Volume 76 No. 4 2012, 529-548 AP ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu ijpam.eu AN OVERVIEW OF SEPARATION AXIOMS IN RECENT RESEARCH D. Narasimhan Department of Mathematics Srinivasa Ramanujan Centre Sastra University Thanjavur, INDIA Abstract: The aim of this paper is to exhibit the recent research on sepa- ration axioms, T -space, pairwise T -space, semi star generalized W -T 1 space, S S 2 pairwise semi star generalized W -T 1 spaces and pairwise complemented spaces 2 and its properties. AMS Subject Classification: 54E55 Key Words: T -space, pairwise T -space, semi star generalized W -T 1 space, S S 2 pairwise semi star generalized W -T 1 spaces 2 1. Introduction Separation axioms are one among the most common, important and interest- ing concepts in Topology. They can be used to define more restricted classes of topological spaces. The separation axioms of topological spaces are usually denoted with the letter “T” after the German “Trennung” which means sepa- ration. Most of the weak separation axioms are defined in terms of generalized closed sets and their definitions are deceptively simple. However, the structure and the properties of those spaces are not always that easy to comprehend. The separation axioms that were studied together in this way were the axioms for Hausdorff spaces, regular spaces and normal spaces. Separation c 2012 Academic Publications, Ltd. Received: December 20, 2011 url: www.acadpubl.eu 530 D. Narasimhan axioms and closed sets in topological spaces have been very useful in the study of certain objects in digital topology [1, 2].
    [Show full text]
  • Separation Axiom
    SEPARATION AXIOM DISCLAIMER Note that in this context the word axiom is not used in the meaning of "principle" of a theory, which has necessarily to be assumed, but in the meaning of "requirement" contained in a definition, which can be fulfilled or not, depending on the cases. Notation Circle – open sets Rectangle with sharp edges – topological space Rectangle with blunt edges – closed sets T0-separation axiom Kolmogorov space For any two points x , y in X, there is an open set U such that x in U and y not in U or y in U and x not in U. X U x y Example of To-space • Every discrete topological space is To – space , for if x and y are two distinct points of X, then {x} contains x and does not contain y. • A non-discrete space containing more than one point is not To – space. • An indiscrete space is not To • A cofinite topological space such that X is an infinite set for if x and y are two distinct points of X, then{x} being finite, X-{x} is open set containing x but not y. T1-separation axiom Frechet spaces For any two points x, y in X there exists two open sets U and V such that x in U and y not in U, and y in V and x not in V. X U V x y Example of T1-space • Every discrete topological space with two or more points is T1 – space , for if x and y are two distinct points of X, then {x} contains x and does not contain y and {y} contains y that does not contain x.
    [Show full text]
  • Complete Regularity As a Separation Axiom
    COMPLETE REGULARITY AS A SEPARATION AXIOM J. DE GROOT AND J. M. AARTS 1. Introduction. Although the axiom of complete regularity ought to be a separation axiom, in none of its usual forms does it look like an intrinsic separation axiom. Our aim in this paper is to establish such characterizations of complete regularity which naturally fit in between regularity and normality and which already have proved to be fundamental and useful. This can simply be achieved by replacing the family of all open sets (as used in the formulation of the separation axioms) by some suitable (sub)base of open sets. For the sake of simplicity, we assume all our spaces to be 7\ and we shall operate with (sub)bases of closed sets (instead of open sets). It is appropriate to define the notion of a screening. Two subsets A and B of a set X are said to be screened by the pair (C, D) if CU D = X,A r\D = 0 and C C\ B = 0. (Consequently, A C C and BCD.) Then we have the following result. THEOREM 1. A space X is completely regular if and only if there is a base S3 for the closed subsets of X such that 1. {Base-regularity) If B Ç S3 and x ÇL B, then {%} and B are screened by a pair from S3; 2. (Base-normality) Every two disjoint elements of S3 are screened by a pair from S3. Two subsets A and B of a space X are said to be screened by a finite family S if 6 covers X and each element of 6 meets1 at most one of A and B.
    [Show full text]
  • A Definition of Separation Axiom
    Canad. Math. Bull. Vol. 17 (4), 1974 A DEFINITION OF SEPARATION AXIOM BY T. CRAMER 0. Introduction. Several separation axioms, defined in terms of continuous functions, were examined by van Est and Freudenthal [3], in 1951. Since that time, a number of new topological properties which were called separation axioms were defined by Aull and Thron [1], and later by Robinson and Wu [2], This paper gives a general definition of separation axiom, defined in terms of continuous functions, and shows that the standard separation axioms, and all but one of these new topological properties, fit this definition. Moreover, it is shown that the re­ maining property, defined in [2], can never fit the expected form of the definition. In addition, a new class of separation axioms lying strictly between T0 and 7\ are defined and characterized, and examples of spaces satisfying these axioms are produced. 1. The definition. Suppose that s/ and 3S are classes of subsets of topological spaces. Given a class of 2£ of topological spaces, with a distinguished pair of sub­ sets Ax and Bx of each member lef, we define T(s/, 3§,2£) as the class of topological spaces Y such that for every disjoint pair of non-empty subsets A and B of Y with Ass/ and B e 3§, there is a continuous map /: Y-+X with f[A]=Ax and f(B)=Bx> for some lef. By a separation axiom we shall mean a class T(s/9 38, 5T), or the intersection of such classes. Throughout, we shall identify all separation axioms with the class of topological spaces satisfying that axiom.
    [Show full text]