TOPOLOGICAL ASPECTS OF YAO S ROUGH SET
|
|
- Meagan Booth
- 5 years ago
- Views:
Transcription
1 Chapter 5 TOPOLOGICAL ASPECTS OF YAO S ROUGH SET In this chapter, we introduce the concept of transmissing right neighborhood via transmissing expression of a relation R on domain U, and then we study various topological properties of Yao s rough set defined through right neighborhood. 5.1 Introduction Rough set was introduced by Pawlak [44] in 1982 as a tool to deal with vagueness and uncertainty of imprecise data. Considerable amount of works have been done on fundamental results of rough set. Partitioning of a set with an equivalence relation is the core concept behind Pawlak s rough set theory. But this is too restrictive to deal with different real life situations. To handle such type of circumstances, several interesting and meaningful extensions of Pawlak rough set theory like, covering based, relation based and neighborhood based rough sets ([30], [31], [32], [33] etc.) have been introduced where covering or cover (it is a finite family of non empty subsets Contents of this chapter has appeared as a paper entitled Topological Properties of Yao s Rough Set in International Journal of Mathematical and Computer Sciences 7(2), 2011,
2 5.2. BASIC CONCEPTS 57 of the universe whose union is the universe), binary relation and right neighborhood respectively are used instead of the equivalence relation. Y.Y. Yao [64] introduced a new type of rough set based on right neighborhoods and E.F. Lashin et al. [26] have generated a topology for the rough set defined by Yao considering the family S = {xr x U} as subbase for the topology, where R is a binary relation on a finite universe U and xr is the right neighborhood of the element x defined as xr = {y xry}. The family S as the subbase for the induced topology τ will be denoted by S R = {xr x U}. In [29], Z. Li has introduced the concept of transmissing expression of a relation, and obtained several interesting results on topological concepts of rough sets. Topology has its own theory and own significance. Rough set theory combined with topology is expected to provide us with new area to study. 5.2 Basic concepts If a relation is reflexive then the right neighborhood of an element will contain the element. Let us call this right neighborhood as reflexive right neighborhood. We review below the lower and upper approximations as defined in [64] and the notion of transmissing expression of a relation as introduced in [29]. Definition [64] Let U be the universe and R U U be a binary relation. For two elements x, y U, we say that y is R-related to x if xry. Then the set r R (x) = {y U xry} is called the right neighborhood of x U. Definition [64] Let U be the universe and R U U be a binary relation. For any X U, the lower and upper approximations of X are respectively X =
3 5.2. BASIC CONCEPTS 58 r R (x) and X = (X c ) c. If X = X then X said to be an exact set and otherwise r R (x) X a rough set. Example Let U = {a, b, c, d} and R = {(a, b), (a, c), (b, d), (c, a), (d, b)} be a binary relation on U. If X = {a, c}, then X = {a} and X = {a, c}. Definition Let R be a binary relation on X, then (i) R is called a similarity relation on X if R is both reflexive and transitive; (ii) R is called a tolerance relation on X if R is both reflexive and symmetric. Definition [29] Let R and R S be two binary relations on X and A X. For all x, y X, we define xr S y iff xry or { v 1, v 2,..., v n } A such that xrv 1, v 1 Rv 2,..., v n Ry. Then R S is called the transmissing expression of R on A. If R S is the transmissing expression of R on X then R S is called transmissing expression of R. Example In Example , arb and brd ar s d, arc and cra ar s a, brd and drb br s b, cra and arc cr s c, drb and brd dr s d. Here R s is not a transmissing expression of R, since for all x, y U we do not have xr s y. Proposition [29] Let R be a reflexive relation on X and R S be the transmissing expression of R. Then for all x, y X, xr S y if and only if { v 1, v 2,..., v n } X such that xrv 1, v 1 Rv 2,..., v n Ry. We refer to [41] for various concepts of a topological space.
4 5.3. TRANSMISSING RIGHT NEIGHBORHOOD Transmissing right neighborhood In this section, we introduce transmissing right neighborhood of an element by considering a reflexive relation. Then we examine a few topological notions like, compactness, connectedness etc. for the transmissing right neighborhood of an element. Let U be a universal set and R be a reflexive binary relation on A U. Suppose τ R = { X U X = X} is a collection of subsets of U. Then it can be easily shown that (U, τ R ) is a topological space. Example Let U = {a, b, c, d} and R = {(a, a), (b, b), (c, c), (d, d), (a, c), (b, d), (a, d), (c, d)} be a reflexive binary relation on U. Now r R (a) = {a, c, d}, r R (b) = {b, d}, r R (c) = {c, d} and r R (d) = {d}. Then τ R = { X U X = X} = {φ, {d}, {b, d}, {c, d}, {a, c, d}, {b, c, d}, U} is a topology on U and (U, τ R ) is a topological space. Proposition If R is a reflexive relation on U, then for all A U, A = A A c = A c. Proof. Let A = A. It will suffice to prove that A c A c. Let y A c y / A y / r R (x) x such that y r R (x) but r R (x) A r R (x) A x such that y r R (x) A c y r R (x) y A c A c A c. r R (x) A c Proposition Let R be a reflexive relation on U, then the topological space (U, τ R ) has the property that A U is open if and only if A is closed. Proof. A U be an open set A = A A c = A c A c is an open set. A is a closed set. Definition Let R S be the transmissing expression of the reflexive relation R on U. Then the transmissing right neighborhood of x X is defined as r RS (x) = {y U xr S y}.
5 5.3. TRANSMISSING RIGHT NEIGHBORHOOD 60 Lemma Let R be a reflexive relation on U and for each x U, let L x = {y U y r RS (x)}. Then (i) x L x ; (ii) L x τ R ; (iii) {L x } is an open neighborhood base at x ; (iv) L x is compact subset of (U, τ R ); (v) B R = {L x x X} is a base of (U, τ R ). Proof. (i) Since R is a reflexive binary relation, x L x. (ii) It is sufficient to show L x L x. Let y L x. Then { v 1, v 2,..., v n } U such that xrv 1, v 1 Rv 2,..., v n Ry y r R (v n ). For z U, z r R (v n ) v n Rz. So, xrv 1, v 1 Rv 2,..., v n R z xr S z z r (x) z L R S x r R (v n ) L x y r R (v n ) y L x L x L x. r R (v n) L x (iii) Here we prove that for each B τ R and x B, L x B. Let y L x y r (x) y r R S R(x) or { v 1, v 2,..., v n } U such that v 1 r R (x), v 2 r R (v 1 ),..., y r R (v n ). Now x B = B = r R (t), for some t U. r R (t) B If y r R (x), then we claim y B. For otherwise, y B c y B c y r R (m) B c r R (m) for some m U If y is in some right neighborhood of m then that right neighborhood is contained in B c. i.e., y r R (x) r R (x) B c. But xrx x r R (x) and r R (x) B c x B c, which is a contradiction. Hence we have y B. If { v 1, v 2, v 3,..., v n } U such that v 1 r R (x), v 2 r R (v 1 ),..., y r R (v n ), then by the above argument we have, v 1 r R (x) v 1 B; v 2 r R (v 1 ) v 2 B;... ; y r R (v n ) y B.
6 5.3. TRANSMISSING RIGHT NEIGHBORHOOD 61 (iv) Let {G λ λ Λ} be an open covering of L x x G λi for some λ i Λ. Then by (iii) L x G λi. Hence L x is a compact subset of (U, τ R ). (v) Similar to that of (iii). Theorem (U, τ R ) = (U, τ R S ). Proof. From Lemma (vi), we have B R = {L x x X} is a base of (U, τ R ). We need to show that B R = {L x x X} is also a base of (U, τ R ). For this, we have to S show that (i) L x τ R S, and (ii) For any x G τ RS, x L x G. (i) We have r (x) L R S x. Let y L x y r R (x). For each z U, S r RS (x) L x z r (x) z L R S x y r (x) L R S x r (x). R S r RS (x) L x r RS (x) L x (ii) Suppose x G τ. Now for each y L R S x y r R (x). Since x G = S r (x), by (iii) of Lemma 5.3.4, y G, i.e., x L R S x G. r RS (x) G Theorem If (U, τ R ) is a topological space induced by a reflexive relation R, then (i) (U, τ R ) is a first countable space; (ii) (U, τ R ) is a locally compact space. Proof. Follows from (iii) and (v) of Lemma respectively. Theorem If (U, τ R ) is a topological space induced by a tolerance relation R, and R S be the transmissing expression of R, then (i) cl({x}) = L x, where cl({x}) is the closure of {x}; (ii) L x is a connected branch that contains x;
7 5.3. TRANSMISSING RIGHT NEIGHBORHOOD 62 (iii) L x is a separable subset of (U, τ R ). Proof. Let (U, τ R ) be a topological space induced by a tolerance relation R and R S is the transmissing expression of R R S is an equivalence relation. Then L x = {y U y r R S (x)} is an equivalence class for each x U, i.e., L x = [x] R S. (i) Suppose there exists y cl({x}) such that y / L x. Then [x] R S [y] RS = φ for an open neighborhood L y of y, {x} L y = φ y / cl({x}) which contradicts our supposition. Hence we must have y cl({x}) y L x cl({x}) L x. On the other hand, let y L x y [x] R S L x = L y. Suppose G is a neighborhood of y. So, L y G L y G φ L x G φ {x} G φ y cl({x}) L x cl({x}). (ii) First we shall show that L x = {y U y r R (x)} is a connected set. Let A S be a non-empty clopen subset of L x. So, there exists y A L x = [x] R S. Then y L y L y = A [y] R S = [x] RS = L y = L x = A. Hence L x is a connected set. Next, let C x be any connected branch that contains x. We shall show L x = C x. If not, L x is an open and closed proper subset of C x, which contradicts the fact that C x is a connected set. Hence L x = C x. (iii) Form (i) we have {x} is a countable dense subset of L x. Hence L x is a separable subset of (U, τ R ). Theorem If R is a tolerance relation, then (i) (U, τ R ) is a regular space; (ii) (U, τ R ) is a normal space; (iii) (U, τ R ) is a locally connected space; (iv) (U, τ R ) is a locally separable space.
8 5.4. CONCLUSION 63 Proof. (i) Let A be a closed subset of U and x A c. Then A and A c are also two open subsets of U such that A A and x A c. Hence (U, τ R ) is a regular space. (ii) If A and B are two disjoint closed subsets of U then they are also two disjoint open subsets of U. Hence (U, τ R ) is a normal space. (iii) We have from (iii) of Lemma 5.3.4, every open neighborhood of x contains the open neighborhood L x of x which is connected. (iv) Follows from (iii) of Theorem Conclusion Here a study on various topological structures of rough sets is undertaken by introducing transmissing right neighborhood. We have considered the rough set defined by Yao as his definition of rough set is based on the concept of neighborhood, and taken those sets whose lower approximation is equal to itself, to get a topology on the universe of discourse. Then it was found that the set L x, which is the set of the elements belonging to transmissing right neighborhoods of x, is an open set satisfying notions like, compactness and connectedness etc.
Relations and Equivalence Relations
Relations and Equivalence Relations In this section, we shall introduce a formal definition for the notion of a relation on a set. This is something we often take for granted in elementary algebra courses,
More information9 RELATIONS. 9.1 Reflexive, symmetric and transitive relations. MATH Foundations of Pure Mathematics
MATH10111 - Foundations of Pure Mathematics 9 RELATIONS 9.1 Reflexive, symmetric and transitive relations Let A be a set with A. A relation R on A is a subset of A A. For convenience, for x, y A, write
More information1.4 Equivalence Relations and Partitions
24 CHAPTER 1. REVIEW 1.4 Equivalence Relations and Partitions 1.4.1 Equivalence Relations Definition 1.4.1 (Relation) A binary relation or a relation on a set S is a set R of ordered pairs. This is a very
More informationAutomata and Languages
Automata and Languages Prof. Mohamed Hamada Software Engineering Lab. The University of Aizu Japan Mathematical Background Mathematical Background Sets Relations Functions Graphs Proof techniques Sets
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationWeek 4-5: Binary Relations
1 Binary Relations Week 4-5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationWeek 4-5: Binary Relations
1 Binary Relations Week 4-5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More information1 The Local-to-Global Lemma
Point-Set Topology Connectedness: Lecture 2 1 The Local-to-Global Lemma In the world of advanced mathematics, we are often interested in comparing the local properties of a space to its global properties.
More informationMath 730 Homework 6. Austin Mohr. October 14, 2009
Math 730 Homework 6 Austin Mohr October 14, 2009 1 Problem 3A2 Proposition 1.1. If A X, then the family τ of all subsets of X which contain A, together with the empty set φ, is a topology on X. Proof.
More informationNAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
More informationApplications of Some Topological Near Open Sets to Knowledge Discovery
IJACS International Journal of Advanced Computer Science Applications Vol 7 No 1 216 Applications of Some Topological Near Open Sets to Knowledge Discovery A S Salama Tanta University; Shaqra University
More informationON SOME PROPERTIES OF ROUGH APPROXIMATIONS OF SUBRINGS VIA COSETS
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (120 127) 120 ON SOME PROPERTIES OF ROUGH APPROXIMATIONS OF SUBRINGS VIA COSETS Madhavi Reddy Research Scholar, JNIAS Budhabhavan, Hyderabad-500085
More informationWe will begin our study of topology from a set-theoretic point of view. As the subject
p. 1 Math 490-01 Notes 5 Topology We will begin our study of topology from a set-theoretic point of view. As the subject expands, we will encounter various notions from analysis such as compactness, continuity,
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationEconomics 204 Summer/Fall 2017 Lecture 1 Monday July 17, 2017
Economics 04 Summer/Fall 07 Lecture Monday July 7, 07 Section.. Methods of Proof We begin by looking at the notion of proof. What is a proof? Proof has a formal definition in mathematical logic, and a
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationOn minimal models of the Region Connection Calculus
Fundamenta Informaticae 69 (2006) 1 20 1 IOS Press On minimal models of the Region Connection Calculus Lirong Xia State Key Laboratory of Intelligent Technology and Systems Department of Computer Science
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationG. de Cooman E. E. Kerre Universiteit Gent Vakgroep Toegepaste Wiskunde en Informatica
AMPLE FIELDS G. de Cooman E. E. Kerre Universiteit Gent Vakgroep Toegepaste Wiskunde en Informatica In this paper, we study the notion of an ample or complete field, a special case of the well-known fields
More informationLecture 7: Relations
Lecture 7: Relations 1 Relation Relation between two objects signify some connection between them. For example, relation of one person being biological parent of another. If we take any two people at random,
More informationSpring -07 TOPOLOGY III. Conventions
Spring -07 TOPOLOGY III Conventions In the following, a space means a topological space (unless specified otherwise). We usually denote a space by a symbol like X instead of writing, say, (X, τ), and we
More informationThis chapter contains a very bare summary of some basic facts from topology.
Chapter 2 Topological Spaces This chapter contains a very bare summary of some basic facts from topology. 2.1 Definition of Topology A topology O on a set X is a collection of subsets of X satisfying the
More informationSets, Functions and Metric Spaces
Chapter 14 Sets, Functions and Metric Spaces 14.1 Functions and sets 14.1.1 The function concept Definition 14.1 Let us consider two sets A and B whose elements may be any objects whatsoever. Suppose that
More informations P = f(ξ n )(x i x i 1 ). i=1
Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological
More information14 Equivalence Relations
14 Equivalence Relations Tom Lewis Fall Term 2010 Tom Lewis () 14 Equivalence Relations Fall Term 2010 1 / 10 Outline 1 The definition 2 Congruence modulo n 3 Has-the-same-size-as 4 Equivalence classes
More informationChapter 9: Relations Relations
Chapter 9: Relations 9.1 - Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R A B, i.e., R is a set of ordered pairs where the first element from each pair
More informationTopology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng
Topology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng Remark 0.1. This is a solution Manuel to the topology questions of the Topology Geometry
More informationarxiv:math/ v1 [math.lo] 5 Mar 2007
Topological Semantics and Decidability Dmitry Sustretov arxiv:math/0703106v1 [math.lo] 5 Mar 2007 March 6, 2008 Abstract It is well-known that the basic modal logic of all topological spaces is S4. However,
More information{x : P (x)} P (x) = x is a cat
1. Sets, relations and functions. 1.1. Set theory. We assume the reader is familiar with elementary set theory as it is used in mathematics today. Nonetheless, we shall now give a careful treatment of
More informationA NEW LINDELOF SPACE WITH POINTS G δ
A NEW LINDELOF SPACE WITH POINTS G δ ALAN DOW Abstract. We prove that implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ1 which has points G δ. In addition, this space has
More informationAxioms of separation
Axioms of separation These notes discuss the same topic as Sections 31, 32, 33, 34, 35, and also 7, 10 of Munkres book. Some notions (hereditarily normal, perfectly normal, collectionwise normal, monotonically
More informationRough Approach to Fuzzification and Defuzzification in Probability Theory
Rough Approach to Fuzzification and Defuzzification in Probability Theory G. Cattaneo and D. Ciucci Dipartimento di Informatica, Sistemistica e Comunicazione Università di Milano Bicocca, Via Bicocca degli
More informationA NOTE ON Θ-CLOSED SETS AND INVERSE LIMITS
An. Şt. Univ. Ovidius Constanţa Vol. 18(2), 2010, 161 172 A NOTE ON Θ-CLOSED SETS AND INVERSE LIMITS Ivan Lončar Abstract For every Hausdorff space X the space X Θ is introduced. If X is H-closed, then
More informationFilters in Analysis and Topology
Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into
More informationNotas de Aula Grupos Profinitos. Martino Garonzi. Universidade de Brasília. Primeiro semestre 2018
Notas de Aula Grupos Profinitos Martino Garonzi Universidade de Brasília Primeiro semestre 2018 1 Le risposte uccidono le domande. 2 Contents 1 Topology 4 2 Profinite spaces 6 3 Topological groups 10 4
More informationINDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS
INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS JAMES P. KELLY AND JONATHAN MEDDAUGH Abstract. In this paper, we develop a sufficient condition for the inverse limit of upper semi-continuous
More informationFoundations of algebra
Foundations of algebra Equivalence relations - suggested problems - solutions P1: There are several relations that you are familiar with: Relations on R (or any of its subsets): Equality. Symbol: x = y.
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationAn Introduction to Modal Logic V
An Introduction to Modal Logic V Axiomatic Extensions and Classes of Frames Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, November 7 th 2013
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More informationThe Euclidean Topology
Lecture note /topology / lecturer :Zahir Dobeas AL -nafie The Euclidean Topology Introduction In a movie or a novel there are usually a few central characters about whom the plot revolves. In the story
More informationSets and Motivation for Boolean algebra
SET THEORY Basic concepts Notations Subset Algebra of sets The power set Ordered pairs and Cartesian product Relations on sets Types of relations and their properties Relational matrix and the graph of
More informationMath 3T03 - Topology
Math 3T03 - Topology Sang Woo Park April 5, 2018 Contents 1 Introduction to topology 2 1.1 What is topology?.......................... 2 1.2 Set theory............................... 3 2 Functions 4 3
More informationA SECOND COURSE IN GENERAL TOPOLOGY
Heikki Junnila, 2007-8/2014 A SECOND COURSE IN GENERAL TOPOLOGY CHAPTER I COMPLETE REGULARITY 1. Definitions and basic properties..... 3 2. Some examples..... 7 Exercises....9 CHAPTER II CONVERGENCE AND
More informationNeighborhoods Systems: Measure, Probability and Belief Functions
202-207. Neighborhoods Systems: Measure, Probability and Belief Functions T. Y. Lin 1 * tylin@cs.sjsu.edu Y, Y, Yao 2 yyao@flash.lakeheadu.ca 1 Berkeley Initiative in Soft Computing, Department of Electrical
More informationSpaces of continuous functions
Chapter 2 Spaces of continuous functions 2.8 Baire s Category Theorem Recall that a subset A of a metric space (X, d) is dense if for all x X there is a sequence from A converging to x. An equivalent definition
More informationA MATROID EXTENSION RESULT
A MATROID EXTENSION RESULT JAMES OXLEY Abstract. Adding elements to matroids can be fraught with difficulty. In the Vámos matroid V 8, there are four independent sets X 1, X 2, X 3, and X 4 such that (X
More informationEQUIVALENCE RELATIONS (NOTES FOR STUDENTS) 1. RELATIONS
EQUIVALENCE RELATIONS (NOTES FOR STUDENTS) LIOR SILBERMAN Version 1.0 compiled September 9, 2015. 1.1. List of examples. 1. RELATIONS Equality of real numbers: for some x,y R we have x = y. For other pairs
More informationProblems - Section 17-2, 4, 6c, 9, 10, 13, 14; Section 18-1, 3, 4, 6, 8, 10; Section 19-1, 3, 5, 7, 8, 9;
Math 553 - Topology Todd Riggs Assignment 2 Sept 17, 2014 Problems - Section 17-2, 4, 6c, 9, 10, 13, 14; Section 18-1, 3, 4, 6, 8, 10; Section 19-1, 3, 5, 7, 8, 9; 17.2) Show that if A is closed in Y and
More informationMATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1
MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and
More information2 Metric Spaces Definitions Exotic Examples... 3
Contents 1 Vector Spaces and Norms 1 2 Metric Spaces 2 2.1 Definitions.......................................... 2 2.2 Exotic Examples...................................... 3 3 Topologies 4 3.1 Open Sets..........................................
More informationConstructive and Algebraic Methods of the Theory of Rough Sets
Constructive and Algebraic Methods of the Theory of Rough Sets Y.Y. Yao Department of Computer Science, Lakehead University Thunder Bay, Ontario, Canada P7B 5E1 E-mail: yyao@flash.lakeheadu.ca This paper
More informationReal Analysis Chapter 4 Solutions Jonathan Conder
2. Let x, y X and suppose that x y. Then {x} c is open in the cofinite topology and contains y but not x. The cofinite topology on X is therefore T 1. Since X is infinite it contains two distinct points
More informationFunctional Analysis HW #1
Functional Analysis HW #1 Sangchul Lee October 9, 2015 1 Solutions Solution of #1.1. Suppose that X
More informationTHE ANTISYMMETRY BETWEENNESS AXIOM AND HAUSDORFF CONTINUA
http://topology.auburn.edu/tp/ http://topology.nipissingu.ca/tp/ TOPOLOGY PROCEEDINGS Volume 45 (2015) Pages 1-27 E-Published on October xx, 2014 THE ANTISYMMETRY BETWEENNESS AXIOM AND HAUSDORFF CONTINUA
More informationThe Tychonoff Theorem
Requirement for Department of Mathematics Lovely Professional University Punjab, India February 7, 2014 Outline Ordered Sets Requirement for 1 Ordered Sets Outline Ordered Sets Requirement for 1 Ordered
More information3. R = = on Z. R, S, A, T.
6 Relations Let R be a relation on a set A, i.e., a subset of AxA. Notation: xry iff (x, y) R AxA. Recall: A relation need not be a function. Example: The relation R 1 = {(x, y) RxR x 2 + y 2 = 1} is not
More information1. To be a grandfather. Objects of our consideration are people; a person a is associated with a person b if a is a grandfather of b.
20 [161016-1020 ] 3.3 Binary relations In mathematics, as in everyday situations, we often speak about a relationship between objects, which means an idea of two objects being related or associated one
More informationNear approximations via general ordered topological spaces M.Abo-Elhamayel Mathematics Department, Faculty of Science Mansoura University
Near approximations via general ordered topological spaces MAbo-Elhamayel Mathematics Department, Faculty of Science Mansoura University Abstract ough set theory is a new mathematical approach to imperfect
More informationCS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati
CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during
More informationII - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define
1 Measures 1.1 Jordan content in R N II - REAL ANALYSIS Let I be an interval in R. Then its 1-content is defined as c 1 (I) := b a if I is bounded with endpoints a, b. If I is unbounded, we define c 1
More informationWeak Choice Principles and Forcing Axioms
Weak Choice Principles and Forcing Axioms Elizabeth Lauri 1 Introduction Faculty Mentor: David Fernandez Breton Forcing is a technique that was discovered by Cohen in the mid 20th century, and it is particularly
More informationUniquely Universal Sets
Uniquely Universal Sets 1 Uniquely Universal Sets Abstract 1 Arnold W. Miller We say that X Y satisfies the Uniquely Universal property (UU) iff there exists an open set U X Y such that for every open
More informationg 2 (x) (1/3)M 1 = (1/3)(2/3)M.
COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is
More informationInfinite-Dimensional Triangularization
Infinite-Dimensional Triangularization Zachary Mesyan March 11, 2018 Abstract The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector
More informationTopological groups with dense compactly generated subgroups
Applied General Topology c Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 85 89 Topological groups with dense compactly generated subgroups Hiroshi Fujita and Dmitri Shakhmatov Abstract.
More informationThe weak topology of locally convex spaces and the weak-* topology of their duals
The weak topology of locally convex spaces and the weak-* topology of their duals Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Introduction These notes
More informationBlocks and 2-blocks of graph-like spaces
Blocks and 2-blocks of graph-like spaces von Hendrik Heine Masterarbeit vorgelegt der Fakultät für Mathematik, Informatik und Naturwissenschaften der Universität Hamburg im Dezember 2017 Gutachter: Prof.
More informationTOPOLOGY TAKE-HOME CLAY SHONKWILER
TOPOLOGY TAKE-HOME CLAY SHONKWILER 1. The Discrete Topology Let Y = {0, 1} have the discrete topology. Show that for any topological space X the following are equivalent. (a) X has the discrete topology.
More informationLocally Compact Topologically Nil and Monocompact PI-rings
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 1, 179-184. Locally Compact Topologically Nil and Monocompact PI-rings M. I. Ursul Departamentul de Matematică,
More informationSanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle
Topology A Profound Subtitle Dr. Copyright c 2017 Contents I General Topology 1 Compactness of Topological Space............................ 7 1.1 Introduction 7 1.2 Compact Space 7 1.2.1 Compact Space.................................................
More informationMarch 3, The large and small in model theory: What are the amalgamation spectra of. infinitary classes? John T. Baldwin
large and large and March 3, 2015 Characterizing cardinals by L ω1,ω large and L ω1,ω satisfies downward Lowenheim Skolem to ℵ 0 for sentences. It does not satisfy upward Lowenheim Skolem. Definition sentence
More informationA general Stone representation theorem
arxiv:math/0608384v1 [math.lo] 15 Aug 2006 A general Stone representation theorem Mirna; after a paper by A. Jung and P. Sünderhauf and notes by G. Plebanek September 10, 2018 This note contains a Stone-style
More informationInternational Journal of Scientific & Engineering Research, Volume 6, Issue 3, March ISSN
International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 969 Soft Generalized Separation Axioms in Soft Generalized Topological Spaces Jyothis Thomas and Sunil Jacob John
More informationi jand Y U. Let a relation R U U be an
Dependency Through xiomatic pproach On Rough Set Theory Nilaratna Kalia Deptt. Of Mathematics and Computer Science Upendra Nath College, Nalagaja PIN: 757073, Mayurbhanj, Orissa India bstract: The idea
More informationMINIMAL UNIVERSAL METRIC SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 42, 2017, 1019 1064 MINIMAL UNIVERSAL METRIC SPACES Victoriia Bilet, Oleksiy Dovgoshey, Mehmet Küçükaslan and Evgenii Petrov Institute of Applied
More informationInternational Journal of Approximate Reasoning
International Journal of Approximate Reasoning 52 (2011) 231 239 Contents lists available at ScienceDirect International Journal of Approximate Reasoning journal homepage: www.elsevier.com/locate/ijar
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationUltrafilters and Semigroup Algebras
Ultrafilters and Semigroup Algebras Isia T. Dintoe School of Mathematics University of the Witwatersrand (Wits), Johannesburg Supervised by: Prof. Yuliya Zelenyuk 31 August 2015 Submitted in partial fulfilment
More informationCSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Determine whether a relation is an equivalence relation by determining whether it is Reflexive Symmetric
More informationTheory of Computation
Thomas Zeugmann Hokkaido University Laboratory for Algorithmics http://www-alg.ist.hokudai.ac.jp/ thomas/toc/ Lecture 1: Introducing Formal Languages Motivation I This course is about the study of a fascinating
More informationTheorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers
Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower
More informationSOLUTIONS TO THE FINAL EXAM
SOLUTIONS TO THE FINAL EXAM Short questions 1 point each) Give a brief definition for each of the following six concepts: 1) normal for topological spaces) 2) path connected 3) homeomorphism 4) covering
More informationLecture 8: Equivalence Relations
Lecture 8: Equivalence Relations 1 Equivalence Relations Next interesting relation we will study is equivalence relation. Definition 1.1 (Equivalence Relation). Let A be a set and let be a relation on
More informationMath 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together
More informationTechnical Results on Regular Preferences and Demand
Division of the Humanities and Social Sciences Technical Results on Regular Preferences and Demand KC Border Revised Fall 2011; Winter 2017 Preferences For the purposes of this note, a preference relation
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationDef. A topological space X is disconnected if it admits a non-trivial splitting: (We ll abbreviate disjoint union of two subsets A and B meaning A B =
CONNECTEDNESS-Notes Def. A topological space X is disconnected if it admits a non-trivial splitting: X = A B, A B =, A, B open in X, and non-empty. (We ll abbreviate disjoint union of two subsets A and
More informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationDENSELY k-separable COMPACTA ARE DENSELY SEPARABLE
DENSELY k-separable COMPACTA ARE DENSELY SEPARABLE ALAN DOW AND ISTVÁN JUHÁSZ Abstract. A space has σ-compact tightness if the closures of σ-compact subsets determines the topology. We consider a dense
More informationRough Soft Sets: A novel Approach
International Journal of Computational pplied Mathematics. ISSN 1819-4966 Volume 12, Number 2 (2017), pp. 537-543 Research India Publications http://www.ripublication.com Rough Soft Sets: novel pproach
More informationReal Analysis. Joe Patten August 12, 2018
Real Analysis Joe Patten August 12, 2018 1 Relations and Functions 1.1 Relations A (binary) relation, R, from set A to set B is a subset of A B. Since R is a subset of A B, it is a set of ordered pairs.
More informationThus, X is connected by Problem 4. Case 3: X = (a, b]. This case is analogous to Case 2. Case 4: X = (a, b). Choose ε < b a
Solutions to Homework #6 1. Complete the proof of the backwards direction of Theorem 12.2 from class (which asserts the any interval in R is connected). Solution: Let X R be a closed interval. Case 1:
More informationCSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Determine whether a relation is an equivalence relation by determining whether it is Reflexive Symmetric
More informationCHAPTER 5. The Topology of R. 1. Open and Closed Sets
CHAPTER 5 The Topology of R 1. Open and Closed Sets DEFINITION 5.1. A set G Ω R is open if for every x 2 G there is an " > 0 such that (x ", x + ") Ω G. A set F Ω R is closed if F c is open. The idea is
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationProperties of the Integers
Properties of the Integers The set of all integers is the set and the subset of Z given by Z = {, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, }, N = {0, 1, 2, 3, 4, }, is the set of nonnegative integers (also called
More informationALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30
ALGEBRAIC GEOMETRY (NMAG401) JAN ŠŤOVÍČEK Contents 1. Affine varieties 1 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 1. Affine varieties The basic objects
More informationOn nano π g-closed sets
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 4 (2018), pp. 611 618 Research India Publications http://www.ripublication.com/gjpam.htm On nano π g-closed sets P. Jeyalakshmi
More information