Equation (2.3) Suggests the Following Model for the Probability of Having a Change-Point Among the First N - 1 Observations

Total Page:16

File Type:pdf, Size:1020Kb

Equation (2.3) Suggests the Following Model for the Probability of Having a Change-Point Among the First N - 1 Observations INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, same thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy subrnitted. Broken or indistinct print, coîored or poor quality illustrations and photographs, pnnt bleedthrough, substandard margins. and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthanzed copyright material had to be removed, a note will indicate the deletion. Oversize materials (e-g., maps, drawings, &arts) are reproduced by sectioning the original, beginning at the upper left-hand merand continuing from left to right in equal sections with small overlaps. Each original is aldo photographed in one exposure and is induded in reduced forrn at the back of the book. Photographs induded in the original manuscript have been repraduced xerographically in this copy. Higher quality 6' x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional diarge. Contact UMI diredly to order. Bell 8 Howell Information and Leaming 300 North Zeeb Road, Ann Arbor, MI 48106-1346 USA 800-521-0600 NOTE TO USERS Page(s) not included in the original manuscript are unavailable from the author or university. The manuscript was microfilmed as received. This reproduction is the best copy available. UMI Modeling Covariance in Multi-Path Changepoint Problems Masoud Asgharian Dastenaei Depart ment of Mat hematics and Statistics McGill University, Montreal A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements of the degree of Doctor of Philosophy OMasoud Asgharian Dastenaei 1998 i National library Bibliothèque nationale 1+1 of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395, nre Wellington Ottawa ON KIA ON4 OrtawaON KlAûN4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, dismbuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/nlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othewise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. To the rnemory of my mother, Hagar, who constantly supported me up to the last day of her life and loved to see this moment, but it didn't corne true. And to my wife and dear friend Mojgan Bien qu'il ait été intensivement étudié dans le cas de la trajectoire unique, le problème du point de changement a été largement ignoré dans le cas de trajectoires multiples. Dans la situation "multi-trajectoires" , il est souvent utile de déterminer l'impact des covariables sur le point de changement lui-même aussi bien que les paramètres avant et après celui-ci. Cette thèse aborde l'inclusion des covariables dans la distribution du point de changement, cet aspect n'ayant jamais été étudié aupara- vant. Le modèle que nous introduisons est basé sur la fonction hasard du change- ment. Il a des caractéristiques qui permettent d'établir les résultats asymptotiques nécessaires à l'estimation et aux tests. En effet, nous établissons la consistance des estimateurs du maximum de vraissemblance des paramètres de notre modèle. Le modèle proposé étant un mélange, deux difficultés reliées à de tels modèles sont à surmonter, à savoir l'identifiabilité et la définie positivité de la matrice d'information. 11 est établi, sous des conditions appropriées, que l'ensemble des zéros du déterminant de la matrice d'information est dense nulle part, paliant ainsi à l'impossibilité d'une preuve directe de la définie positivité. En utilisant la méthode d'annulation par simuation, nous avons effectué quelques simulations afin de déterminer la maniabilité de notre procédure d'estimation. Dans l'exemple traité, notre estimateur semble approximativement suivre une nor- male, même pour des échantillons de taille modérée. Les estimateurs du maximum de vraissemblance semblent également bien approximer leur paramètres. ABSTRACT Although the single-path changepoint problem has been extensively treated in the statistical literature, the multi-path changepoint problem has been largely ignored. In the multi-path changepoint setting it is often of interest to assess the impact of covariates on the changepoint itself as weil as on the parameters before and after the changepoint. This thesis is concerned with including covariates in the changepoint distribution, a topic never before addressed in the literature. The model we introduce, based on the hazard of change, enjoys features which allow one to establish asymptotic results needed for estimation and testing. Indeed, we establish consistency of the maximum likelihood estimators of the parameters of Our model. -4s the proposed model is a mixture model, two of the difficulties associated with such models are addressed. They are identifiability, and positive definiteness of the information matrix. It is shown that under suitable conditions the set of zeros of the determinant of the information matrix is a nowhere dense set: thus partially compensating for the impossibility of directly establishing positive definiteness. A limited simulation, using simulated annealing, is carried out to assess how the estimation procedure works in practice. In the esample presented, the estimators appear to follow an approximately normal distribution even for moderate sample sizes. The maximum likelihood estirnators appear to approximate their parameter counterparts well. Chapter 4: 93 : Lemma 1. Lemma 2. Lemma 3 and Theorem 5, Lemma 5. Lemma 6. Proposition 4. Theorem 7. Lemma 7, and Theorem 8 $4 : Lemma 8. Theorem 9 and Theorem 10 + $3 : Establishing asym ptot ic normality of the maximum likelihood estimators of the unknown parameters in the mode1 introduced in Chapter 1 ACKNOWLEGEMENT David Wolfson has been much more than a supervisor for me. He has been a mise friend. His fastidiousness enorrnously improved the exposition of this thesis and his unusual patience gave me the chance to work on a varîety of problems and enjoy learning new things. He and his wife, Tina Wolfson of the Division of Clinical Epidemiology at the Jewish General Hospital(JGH), provided me the chance to learn about aspects of statistics not covered in the classroom. Indeed, working at the Jewish General Hospital forced me to understand many things which 1 had never questioned before. It was at the JGH that I was given the chance to work on survivai analysis, my favourite topic in statistics, and where 1 was introduced to the notion of length-biased sampling. For al1 this 1 would like to express my sincere gratitude. 1 thank Sanjo Zlobec for interesting lectures on parametric programrning and inspiration for working on an ongoing problem. Jal Choksi always gave me the most relevant references to my questions. -4mong my friends 1 should start with Enrique Reyes whorn 1 tortured pitilessly with my questions on differential geometry. He introduced me to the book by -4braham, Marsden and Ratiu(1988) which turned out to be my main reference in Chapter 3 of this thesis. Luc Lalond was a source of computer skills from which 1 personally benefitted very much. When 1was stuck for a long time with an error in my program, he devoted a considerable amount of time to find the error, although he was very busy himself. Statistical discussion with my old friend Khalil Shafie has been always beneficial for me. 1 also befitted very much from his computer skills. 1 also like to thank Lassina Dembele who helped me with translation of the abstract of the thesis. 1 am very grateful to the Ministry of Higher Education of Iran for supporting me through my education. 1 would also like to express my acknowledgrnent to the McGill Major Fellowship Foundation who awarded me the "175th Anniversary of McGill University" fellowship . When, for some baseless reason 1 was labekd as somebody who neither has the right to continue his education abroad, nor even in Iran, it was my Masters supervisor, Siamak Xoorbaloochi, who helped me overcome this obstacle. Without him there would not be any thesis nor even an education towards a PhD. 1 am most indebted to him for al1 he did for me. My siblings have always been a great source of encouragement and inspiration. 1 don't think there are any words that can express my real gratitude and acknowl- edgrnent to them. 1 am also very grateful to my father and mother-in-law who helped me and my wife very much. Contents Chapter 1. INTRODUCTION 3 Chapter II. HAZARD APPROACH IN THE MULTI-PATH CHANGE POINT 7 1. Introduction 2. Markovian Structure Of The Changepoint Problem 3. Principle of Maximum Entropy and Modeling 22 3.1. Synopsis Of History And Etymology Of The Word " Entropy" 22 3.2. Entropy And Slodeling 24 4. Introducing Covariates into the Model 5. Mixture Distributions Chapter III. CONSISTENCY OF THE MLE 1. Introduction 51 2. Identifiability Of The Mode1 52 3. Consistency In The Single Parameter Case 61 4. Consistency In The Multiparameter Case For Identifiable Mod- els 72 1 2 CONTENTS 5.
Recommended publications
  • The Continuity of Additive and Convex Functions, Which Are Upper Bounded
    THE CONTINUITY OF ADDITIVE AND CONVEX FUNCTIONS, WHICH ARE UPPER BOUNDED ON NON-FLAT CONTINUA IN Rn TARAS BANAKH, ELIZA JABLO NSKA,´ WOJCIECH JABLO NSKI´ Abstract. We prove that for a continuum K ⊂ Rn the sum K+n of n copies of K has non-empty interior in Rn if and only if K is not flat in the sense that the affine hull of K coincides with Rn. Moreover, if K is locally connected and each non-empty open subset of K is not flat, then for any (analytic) non-meager subset A ⊂ K the sum A+n of n copies of A is not meager in Rn (and then the sum A+2n of 2n copies of the analytic set A has non-empty interior in Rn and the set (A − A)+n is a neighborhood of zero in Rn). This implies that a mid-convex function f : D → R, defined on an open convex subset D ⊂ Rn is continuous if it is upper bounded on some non-flat continuum in D or on a non-meager analytic subset of a locally connected nowhere flat subset of D. Let X be a linear topological space over the field of real numbers. A function f : X → R is called additive if f(x + y)= f(x)+ f(y) for all x, y ∈ X. R x+y f(x)+f(y) A function f : D → defined on a convex subset D ⊂ X is called mid-convex if f 2 ≤ 2 for all x, y ∈ D. Many classical results concerning additive or mid-convex functions state that the boundedness of such functions on “sufficiently large” sets implies their continuity.
    [Show full text]
  • Math 501. Homework 2 September 15, 2009 ¶ 1. Prove Or
    Math 501. Homework 2 September 15, 2009 ¶ 1. Prove or provide a counterexample: (a) If A ⊂ B, then A ⊂ B. (b) A ∪ B = A ∪ B (c) A ∩ B = A ∩ B S S (d) i∈I Ai = i∈I Ai T T (e) i∈I Ai = i∈I Ai Solution. (a) True. (b) True. Let x be in A ∪ B. Then x is in A or in B, or in both. If x is in A, then B(x, r) ∩ A , ∅ for all r > 0, and so B(x, r) ∩ (A ∪ B) , ∅ for all r > 0; that is, x is in A ∪ B. For the reverse containment, note that A ∪ B is a closed set that contains A ∪ B, there fore, it must contain the closure of A ∪ B. (c) False. Take A = (0, 1), B = (1, 2) in R. (d) False. Take An = [1/n, 1 − 1/n], n = 2, 3, ··· , in R. (e) False. ¶ 2. Let Y be a subset of a metric space X. Prove that for any subset S of Y, the closure of S in Y coincides with S ∩ Y, where S is the closure of S in X. ¶ 3. (a) If x1, x2, x3, ··· and y1, y2, y3, ··· both converge to x, then the sequence x1, y1, x2, y2, x3, y3, ··· also converges to x. (b) If x1, x2, x3, ··· converges to x and σ : N → N is a bijection, then xσ(1), xσ(2), xσ(3), ··· also converges to x. (c) If {xn} is a sequence that does not converge to y, then there is an open ball B(y, r) and a subsequence of {xn} outside B(y, r).
    [Show full text]
  • Baire Category Theorem
    Baire Category Theorem Alana Liteanu June 2, 2014 Abstract The notion of category stems from countability. The subsets of metric spaces are divided into two categories: first category and second category. Subsets of the first category can be thought of as small, and subsets of category two could be thought of as large, since it is usual that asset of the first category is a subset of some second category set; the verse inclusion never holds. Recall that a metric space is defined as a set with a distance function. Because this is the sole requirement on the set, the notion of category is versatile, and can be applied to various metric spaces, as is observed in Euclidian spaces, function spaces and sequence spaces. However, the Baire category theorem is used as a method of proving existence [1]. Contents 1 Definitions 1 2 A Proof of the Baire Category Theorem 3 3 The Versatility of the Baire Category Theorem 5 4 The Baire Category Theorem in the Metric Space 10 5 References 11 1 Definitions Definition 1.1: Limit Point.If A is a subset of X, then x 2 X is a limit point of X if each neighborhood of x contains a point of A distinct from x. [6] Definition 1.2: Dense Set. As with metric spaces, a subset D of a topological space X is dense in A if A ⊂ D¯. D is dense in A. A set D is dense if and only if there is some 1 point of D in each nonempty open set of X.
    [Show full text]
  • Real Analysis, Math 821. Instructor: Dmitry Ryabogin Assignment
    Real Analysis, Math 821. Instructor: Dmitry Ryabogin Assignment IV. 1. Problem 1. Definition 1. A set M ⊂ R is called closed if it coincides with its closure. In other words, M ⊂ R is closed, if it contains all of its limit points. Definition 2. A set A ⊂ R is called open if there is a closed set M ⊂ R such that A = R n M. In other words, A is open if all of its points are inner points, (a point x is called the inner point of a set A, if there exists > 0, such that the -neighbourhood U(x) of x, is contained in A, U(x) ⊂ A). Prove that any open set on the real line is a finite or infinite union of disjoint open intervals. (The sets (−∞; 1), (α, 1), (−∞; β) are intervals for us). Hint. Let G be open on R. Introduce in G the equivalence relationship, by saying that x ∼ y if there exists an interval (α, β), such that x; y 2 (α, β) ⊂ G. a) Prove at first that this is indeed the equivalence relationship, i. e., that x ∼ x, x ∼ y implies y ∼ x, and x ∼ y, y ∼ z imply x ∼ z. Conclude that G can be written as a disjoint union G = [τ2iIτ of \classes" of points Iτ , Iτ = fx 2 G : x ∼ τg, i is the set of different indices, i.e. τ 2 i if Iτ \ Ix = ?, 8x 2 G. b) Prove that each Iτ is an open interval. c) Show that there are at most countably many Iτ (the set i is countable).
    [Show full text]
  • Arxiv:1809.06453V3 [Math.CA]
    NO FUNCTIONS CONTINUOUS ONLY AT POINTS IN A COUNTABLE DENSE SET CESAR E. SILVA AND YUXIN WU Abstract. We give a short proof that if a function is continuous on a count- able dense set, then it is continuous on an uncountable set. This is done for functions defined on nonempty complete metric spaces without isolated points, and the argument only uses that Cauchy sequences converge. We discuss how this theorem is a direct consequence of the Baire category theorem, and also discuss Volterra’s theorem and the history of this problem. We give a sim- ple example, for each complete metric space without isolated points and each countable subset, of a real-valued function that is discontinuous only on that subset. 1. Introduction. A function on the real numbers that is continuous only at 0 is given by f(x) = xD(x), where D(x) is Dirichlet’s function (i.e., the indicator function of the set of rational numbers). From this one can construct examples of functions continuous at only finitely many points, or only at integer points. It is also possible to construct a function that is continuous only on the set of irrationals; a well-known example is Thomae’s function, also called the generalized Dirichlet function—and see also the example at the end. The question arrises whether there is a function defined on the real numbers that is continuous only on the set of ra- tional numbers (i.e., continuous at each rational number and discontinuous at each irrational), and the answer has long been known to be no.
    [Show full text]
  • Arxiv:1603.05121V3 [Math.GM] 31 Oct 2016 Definition of Set Derived Efc (I.E
    SOME EXISTENCE RESULTS ON CANTOR SETS Borys Alvarez-Samaniego´ N´ucleo de Investigadores Cient´ıficos Facultad de Ingenier´ıa, Ciencias F´ısicas y Matem´atica Universidad Central del Ecuador (UCE) Quito, Ecuador Wilson P. Alvarez-Samaniego´ N´ucleo de Investigadores Cient´ıficos Facultad de Ingenier´ıa, Ciencias F´ısicas y Matem´atica Universidad Central del Ecuador (UCE) Quito, Ecuador Jonathan Ortiz-Castro Facultad de Ciencias Escuela Polit´ecnica Nacional (EPN) Quito, Ecuador Abstract. The existence of two different Cantor sets, one of them contained in the set of Liouville numbers and the other one inside the set of Diophan- tine numbers, is proved. Finally, a necessary and sufficient condition for the existence of a Cantor set contained in a subset of the real line is given. MSC: 54A05; 54B05 Keywords: Cantor set; Liouville numbers; Diophantine numbers. arXiv:1603.05121v3 [math.GM] 31 Oct 2016 1. Introduction First, we will introduce some basic topological concepts. Definition 1.1. A nowhere dense set X in a topological space is a set whose closure has empty interior, i.e. int(X)= ∅. Definition 1.2. A nonempty set C ⊂ R is a Cantor set if C is nowhere dense and perfect (i.e. C = C′, where C′ := {p ∈ R; p is an accumulation point of C} is the derived set of C). Definition 1.3. A condensation point t of a subset A of a topological space, is any point t, such that every open neighborhood of t contains uncountably many points of A. Date: October 31, 2016. 1 2 B. ALVAREZ-SAMANIEGO,´ W.
    [Show full text]
  • Planetmath: Topological Space
    (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search topological space (Definition) "topological space" is owned by djao. [ full author list (2) ] (more info) Math for the people, by the people. Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS Advanced search compact (Definition) "compact" is owned by djao. [ full author list (2) ] Dense set 1 Dense set In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A - for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation). Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty. The density of a topological space X is the least cardinality of a dense subset of X. Density in metric spaces An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points), Then A is dense in X if Note that .
    [Show full text]
  • Structured Linearizations for Matrix Polynomials
    Structured Linearizations for Matrix Polynomials Mackey, D. Steven 2006 MIMS EPrint: 2006.68 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 STRUCTURED LINEARIZATIONS FOR MATRIX POLYNOMIALS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2006 D. Steven Mackey School of Mathematics Contents Abstract 6 Declaration 7 Copyright 8 Publications 9 Advisor and Examiners 10 Acknowledgements 11 1 Introduction 13 1.1 Some Preliminaries . 16 2 A Vector Space Setting for Linearizations 19 2.1 Vector Spaces of \Potential" Linearizations . 19 2.2 When is a Pencil in L1(P ) a Linearization? . 25 2.2.1 The Strong Linearization Theorem . 25 2.2.2 Linearization Conditions . 29 2.3 Another View of L1(P ) and L2(P ) ................... 31 3 DL(P ) and Block-symmetry 33 3.1 The Block-transpose Operation . 33 3.2 Block-symmetry and Shifted Sum Equations . 36 3.2.1 Shifted Sum Equations . 37 3.3 Block-symmetric Pencils in L1(P ).................... 39 3.3.1 The Subspace B(P )........................ 40 3.3.2 The \Standard Basis" for B(P ) . 40 3.4 Double Ansatz Pencils for General P . 44 3.5 Other Block-symmetric Linearizations . 45 4 The Genericity of Linearizations in DL(P ) 48 4.1 Some Suggestive Examples . 48 4.2 Determinant of DL(P )-pencils .
    [Show full text]
  • On Spaces Whose Nowhere Dense Subsets Are Scati'ered
    Internat. J. Math. & Math. Sci. 735 VOL. 21 NO. 4 (1998) 735-740 ON SPACES WHOSE NOWHERE DENSE SUBSETS ARE SCATI'ERED JULIAN DONTCHEV DAVID ROSE Department ofMathematics Department ofMathematics University of Helsinki Southeastern College Hallituskatu 15 1000 Longfellow Boulevard 00014 Helsinki 0, FINLAND Lakeland, Florida 33801-6099, USA (Received November 12, 1996) ABSTRACT. The am of this paper is to study the class of N-scattered spaces, i.e. the spaces whose nowhere dense subsets are scattered. The concept was recently used in a decomposition of scatteredness a topological space (X, 7") is scattered if and only if X is a-scattered its a-topology is scattered) and N-scattered. KEY WORDS AND PHRASES: Scattered space, a-scattered, N-scattered, rim-scattered, a-space, topological ideal. 1991 AMS SUBJECT CLASSHClCATION CODES: Primary: 54G12, 54G05; Secondary: 54G15, 54G99. 1. INTRODUCTION A topological space (X, 7-) is scattered if every nonempty subset of X has an isolated point, e. if X has no nonempty dense-in-itself subspace. The a-topology on X denoted by r [6] is the collection of all subsets of the form U\N, where U is open and N is nowhere dense or equivalently all sets A satisfying A C_ Int IntA. If r 7-, then X is said to be an a-space or a nodec space [5]. All subrnaximal and all globally disconnected spaces are examples of c-spaees. We recall that a space X is submaximal if every dense set is open and globally disconnected if every set which can be placed between an open set and its closure is open [2].
    [Show full text]
  • Measure and Category
    Measure and Category Marianna Cs¨ornyei [email protected] http:/www.ucl.ac.uk/∼ucahmcs 1 / 96 A (very short) Introduction to Cardinals I The cardinality of a set A is equal to the cardinality of a set B, denoted |A| = |B|, if there exists a bijection from A to B. I A countable set A is an infinite set that has the same cardinality as the set of natural numbers N. That is, the elements of the set can be listed in a sequence A = {a1, a2, a3,... }. If an infinite set is not countable, we say it is uncountable. I The cardinality of the set of real numbers R is called continuum. 2 / 96 Examples of Countable Sets I The set of integers Z = {0, 1, −1, 2, −2, 3, −3,... } is countable. I The set of rationals Q is countable. For each positive integer k there are only a finite number of p rational numbers q in reduced form for which |p| + q = k. List those for which k = 1, then those for which k = 2, and so on: 0 1 −1 2 −2 1 −1 1 −1 = , , , , , , , , ,... Q 1 1 1 1 1 2 2 3 3 I Countable union of countable sets is countable. This follows from the fact that N can be decomposed as the union of countable many sequences: 1, 2, 4, 8, 16,... 3, 6, 12, 24,... 5, 10, 20, 40,... 7, 14, 28, 56,... 3 / 96 Cantor Theorem Theorem (Cantor) For any sequence of real numbers x1, x2, x3,..
    [Show full text]
  • Dense Sets, Nowhere Dense Sets and an Ideal in Generalized Closure Spaces
    MATEMATIQKI VESNIK UDK 515.122 59 (2007), 181–188 originalni nauqni rad research paper DENSE SETS, NOWHERE DENSE SETS AND AN IDEAL IN GENERALIZED CLOSURE SPACES Chandan Chattopadhyay Abstract. In this paper, concepts of various forms of dense sets and nowhere dense sets in generalized closure spaces have been introduced. The interrelationship among the various notions have been studied in detail. Also, the existence of an ideal in generalized closure spaces has been settled. 1. Introduction Structure of closure spaces is more general than that of topological spaces. Hammer studied closure spaces extensively in [8,9], and a recent study on these spaces can be found in Gnilka [5,6], Stadler [14,15], Harris [10], Habil and Elzena- ti [7]. Although the applications of general topology is not available so much in digital topology, image analysis and pattern recognition; the theory of generalized closure spaces has been found very important and useful in the study of image analysis [3,13]. In [14,15], Stadler studied separation axioms on generalized closure spaces. The following definition of a generalized closure space can be found in [7] and [15]. Let X be a set. }(X) be its power set and cl : }(X) ¡! }(X) be any arbitrary set-valued set function, called a closure function. We call clA, A ½ X, the closure of A and we call the pair (X; cl) a generalized closure space. The closure function in a generalized closure space (X; cl) is called: (a) grounded if cl(;) = ;, (b) isotonic if A ½ B ) clA ½ clB, (c) expanding if A ½ clA for all A ½ X, (d) sub-additive if cl(A [ B) ½ clA [ clB, (e) idempotent if cl(clA) = clA, S S (f) additive if ¸2Λ cl(A¸) = cl( ¸2Λ(A¸)).
    [Show full text]
  • Generalized Nowhere Dense Sets in Cluster Topological Setting
    International Journal of Pure and Applied Mathematics Volume 109 No. 2 2016, 459-467 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v109i2.19 ijpam.eu GENERALIZED NOWHERE DENSE SETS IN CLUSTER TOPOLOGICAL SETTING M. Matejdes Department of Mathematics and Computer Science Faculty of Education Trnava University in Trnava Priemyseln´a4, 918 43 Trnava, SLOVAKIA Abstract: The aim of the article is to generalize the notion of nowhere dense set with respect to a cluster topological space which is defined as a triplet (X, τ, E) where (X, τ) is a topological space and E is a nonempty family of nonempty subsets of X. The notions of E-nowhere dense and locally E-scattered sets are introduced and the necessary and sufficient conditions under which the family of all E-nowhere dense sets is an ideal are given. AMS Subject Classification: 54A05, 54E52, 54G12 Key Words: cluster system, cluster topological space, ideal topological space, E-scattered set, E-nowhere dense set 1. Introduction and Basic Definitions Cluster topological spaces provide a general framework with the involvement of ideal topological spaces [1], [2], [3], [9]. They have a wider application and its progress can find in [6], [7], [10]. The paper can be considered as a con- tinuation of [5] where some cluster topological notions were introduced and it corresponds with the efforts to generalize the Baire classification of sets and the Baire category theorem [4], [11]. In [5] one can find an open problem to discover a necessary and sufficient condition under which the family of all E-nowhere dense sets forms an ideal.
    [Show full text]