Lecture : Topological Space
|
|
- Ethelbert Leonard
- 5 years ago
- Views:
Transcription
1 Example of Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014
2 Outline Example of Example of 4 5 6
3 Example of I Topological spaces and continuous functions between them are the primary objects of study in the field of topology. In next 1 to 15 lectures, we introduce topological spaces and some important concepts associated with them, including open sets, bases, and closed sets, clouter and interior. We will present two applications of topological spaces 1 Involving digital image processing 2 Concerning evolutionary proximity in biology In this lecture we will introduce the dedition of and their examples.
4 Example of II For many years, prior to the formalization of the field of topology, mathematicians used the concept of an open set, a simple example of which is an open interval on the real line. But over time it was realized that many of the properties held by open sets on the real line could be said to hold for certain types of subsets in any set. Eventually, the essential properties were distilled out and the concept of a collection of open sets, called a topology, evolved into the definition of topological space.
5 Example of I A topological space is supposed to be a set that has just enough structure to meaningfully speak of continuous functions on it.
6 Example of II Definition Let X be a any non-empty set. A topology on X is a collection τ of subset of X such that: 1 The set X and empty set φ belong to τ. 2 The union of any arbitrary (finite or infinite) number of subsets of τ belongs to τ. In other words if G i belong to τ is arbitrary, then i I G i τ, where I is any set. 3 The intersection of finite many subsets of τ belong to τ. In other words assume that for each i I if G i belong to τ, then i I G i τ, where I is finite set.
7 Example of III In short, a topology on a set X is a collection of subsets of X which includes empty set and X and is closed under unions and finite intersections. The subsets of collection τ are called open. A set together with a topology is called a topological space and denoted by (X, τ). Question: Which components involve to make topological space? There are two things that make up topological space : a set X, and a collection, τ, of subsets of X that forms a topology on X. To be properly formal, we should refer to a topological space as an ordered pair (X, τ), but to simplify notation we follow the common practice of refereing to
8 Example of IV the set X as a topological space, leaving it implicitly understood that there is a topology on X.
9 Example of V Remark 1 The statement intersection of finitely many subsets of τ is equivalent to the statement intersection of two subsets of τ. 2 One should be careful when using the word open. Open intervals in R are generally not the same thing as open sets in a topological space. Open sets are a generalization of the concept of open intervals.
10 Example of Example of I Example If X be the three-point set X = {a, b, c}, the there are many possibilities of topologies on X. Here we considered some of them schematically in figure 1 The set X with topology τ1 = {X, φ}. 2 The set X with topology τ2 = {X, φ, {a}, {a, b}}. 3 The set X with topology τ3 = {X, φ, {b}, {a, b}, {a, c}}. 4 The set X with topology τ4 = {X, φ, {b}}. 5 The set X with topology τ5 = {X, φ, {a}, {b, c}}. 6 The set X with topology τ6 = {X, φ, {b}, {c}, {a, b}, {b, c}}. 7 The set X with topology τ7 = {X, φ, {a, c}}.
11 Example of Example of II 8 The set X with topology τ8 = {X, φ, {a}, {b}, {a, b}}. 9 The set X with topology τ 9 = {X, φ, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}}. On the set of X there are many topologies can be defined but all the collections of subsets of X are not necessary to be topologies on X. The collections {X, φ, {a}, {b}} and {X, φ, {a, b}, {b, c}} of subsets of X are not topologies on X.
12 Example of Example of III Example Let N be the set of all natural numbers (that is, the set of all positive integers) and let τ consist of N, φ and all finite subsets of N. Then collection τ is not a topology on N, because the infinite union {2} {3}... {n}... = {2, 3,..., n,...} of members of τ does not belong to τ.
13 Example of Example of IV We have now defined three different topologies on X; The non-discrete or trivial topology The discrete topology The finite complement topology In the case of these three, the trivial topology, with fewest open sets, is contained within the finite complement topology, which is itself contained within the discrete topology.
14 Example of Example of V Example (Non-discrete Topology) Let X be a non-empty set. Define τ = {φ, X}. Notice that τ satisfies all three of the conditions for being a topology. However, if we remove either set, we no longer have a topology. Thus {φ, X} is minimal topology we can define on X. For obvious reasons, it is called the trivial topology or non-discrete on X.
15 Example of Example of VI Example (Discrete Topology) Let X be a nonempty set and let τ be the collection of all subsets of X. Clearly this is a topology, since unions and intersections of subsets of X are themselves subsets of X and therefore are in the collection τ. We call this the discrete topology on X. This is the largest topology that we can define on X.
16 Example of Example of VII Example (Finite Complement Topology) Let X be any set and τ f is the collections of all subsets U of X such that X U either is finite or empty or all of X i.e. τ f = {U X : X U either finite or empty or all of X} then the collection τ f is topology on X and called finite complement topology on X. Here we can check that above collection is in fact a topology on X. 1 Since X X = φ is finite and X φ = X is all of X, so X and φ belong to τ f.
17 Example of Example of VIII 2 Let {Uα } be a collections of open sets of X i.e. member of τ f,then we want to show that U α is in τ f. And for this we will show that X U α is finite. X U α =X (U 1 U2...) =(X U 1 ) (X U 2 )... Since {U 1, U 2,...} are open so by the definition, {(X U 1 ), (X U 2 ),...} are finite and by the properties of finite set that intersection of finite sets are finite so {(X U 1 ) (X U 2 )...} is finite i.e. X U α is finite. Hence by the hypothesis, U α is in τ f. 3 Similarly, we can check for finite intersection of open sets.
18 Example of Example of IX Example (Complement topology) On the real line R, define a topology whose open sets are the empty set and every set in R with a finite complement. For example, U = R {0, 3, 7} is an open set. We call this topology complement topology on R and denote it by R fc.
19 Example of Example of X Example Consider R, the set of real numbers, with τ = {S R: ɛ > 0 s.t. (x ɛ, x + ɛ) S} Here τ is a topology on R called usual topology and (R, τ) called usual topological space. 1 By the definition of τ, clearly τ P (R). And so we can say that φ τ is vacuously true and also R τ.
20 Example of Example of XI 2 Second, let {Aα }, α I be a family of sets indexed over set I such that for all α I, A α τ. Now we want to show that α I A α τ. Let W = α I A α. For all x in W, there exists α in I such that x in A α. So, by hypothesis, there exists ɛ > 0 such that (x ɛ, x + ɛ) A α W. Therefore, α I A α τ. 3 Finally, let A, B in τ. We must show that A B in τ. By hypothesis, if A and B in τ, then there exists non-zero ɛ a and ɛ b such that (x ɛ a, x + ɛ a ) A and (x ɛ b, x + ɛ b ) B. Now choose ɛ = min{ɛ a, ɛ b }. Then (x ɛ, x + ɛ) A and (x ɛ, x + ɛ) B. So, we can say that (x ɛ, x + ɛ) A B. Therefore, A B in τ.
21 Example of Example of XII Example (Topology induced by the metric) Let (X, d) be a metric space. Define O X to be open if for any x in O, there exists an open ball B(x, r) lying inside O. Then, τ d = {O X : O is open} {φ} is a topology on X. τ d is called the topology induced by the metric d.
22 Example of I Definition Let τ 1 and τ 2 are two topologies on the set X. τ 2 is said to strictly stronger or stronger than τ 1 if τ 2 properly contain τ 1 or τ 2 τ 1 respectively. Definition Similarly, τ 2 is said to strictly weaker or weaker than τ 1 if τ 1 properly contain τ 2 or τ 2 τ 1 respectively. And both topologies are said to be comparable if τ 2 τ 1 and τ 1 τ 2.
23 Example of I Theorem In a topological space (X, τ), an arbitrary union of open sets and a finite intersection of open sets is open. Proof. We can proof this directly by the definition of topological space.
24 Example of II Theorem In a topological space (X, τ), an arbitrary intersection of closed sets and a finite union of closed sets is open. Proof:
25 Example of III Let F i be subset of X for all i in N and G = i=1 F i and H = n i=1 F i. Now F i is closed in X (X F i ) is open (X F i ) and i=1 X n (X F i ) are open sets i=1 F i and X i=1 F i and i=1 Hence G and H are closed sets. n F i are open sets i=1 n F i are closed sets i=1
26 Example of IV Theorem The union of infinite collection of closed sets in a topological space is not necessarily closed. Proof: n Let (R, τ) be topological space and F n = [0, n+1 ] for all n in N be subset of R. Here F n is closed. And F i = i=1 [ 0, 1 ] [ 0, 2 ]... [0, 1) 2 3 = [0, 1) closed set Therefore, i=1 F i is not closed set, even each F n is a closed set.
27 I Example of List all the possible topologies on X = {a, b}.
28 II Example of Let X = {a, b, c, d, e}. Determine whether or not each of the following classes of subsets of the X is a topology on X. 1 τ1 = {X, φ, {a}, {a, b}, {a, c}}. 2 τ2 = {X, φ, {a, b, c}, {a, b, d}, {a, b, c, d}}. 3 τ3 = {X, φ, {a}, {a, b}, {a, c, d}, {a, b, c, d}}.
29 III Example of Prove that (R 2, τ) is a topological space where the elements of τ are φ and the complements of finite sets of lines and points.
30 IV Example of Let τ = {R 2, φ} {G k : k R} be the class of the subsets of the plane R 2 where G k = {(x, y): x, y R, x > y + k} 1 Prove that τ is topology on R 2. 2 If τ is topology on R 2 if k R is replaced by k N, by k Q.
31 V Example of Let τ be the class consists of R, φ all infinite interval A q = (q, ) with q Q, the rationales. Show that τ is not topology on R.
Lecture-12: Closed Sets
and Its Examples Properties of Lecture-12: Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Introduction and Its Examples Properties of 1 Introduction
More informationLecture - 8A: Subbasis of Topology
Lecture - 8A: Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline 1 Introduction 2 3 4 Introduction I As we know that topology generated by a basis B may
More informationTopology 550A Homework 3, Week 3 (Corrections: February 22, 2012)
Topology 550A Homework 3, Week 3 (Corrections: February 22, 2012) Michael Tagare De Guzman January 31, 2012 4A. The Sorgenfrey Line The following material concerns the Sorgenfrey line, E, introduced in
More informationLecture 15: The subspace topology, Closed sets
Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology
More informationTopology Homework 3. Section Section 3.3. Samuel Otten
Topology Homework 3 Section 3.1 - Section 3.3 Samuel Otten 3.1 (1) Proposition. The intersection of finitely many open sets is open and the union of finitely many closed sets is closed. Proof. Note that
More informationOpen and Closed Sets
Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
More informationMath 395: Topology. Bret Benesh (College of Saint Benedict/Saint John s University)
Math 395: Topology Bret Benesh (College of Saint Benedict/Saint John s University) October 30, 2012 ii Contents Acknowledgments v 1 Topological Spaces 1 2 Closed sets and Hausdorff spaces 7 iii iv CONTENTS
More informationMATH 54 - LECTURE 4 DAN CRYTSER
MATH 54 - LECTURE 4 DAN CRYTSER Introduction In this lecture we review properties and examples of bases and subbases. Then we consider ordered sets and the natural order topology that one can lay on an
More information9.5 Equivalence Relations
9.5 Equivalence Relations You know from your early study of fractions that each fraction has many equivalent forms. For example, 2, 2 4, 3 6, 2, 3 6, 5 30,... are all different ways to represent the same
More informationReview of Sets. Review. Philippe B. Laval. Current Semester. Kennesaw State University. Philippe B. Laval (KSU) Sets Current Semester 1 / 16
Review of Sets Review Philippe B. Laval Kennesaw State University Current Semester Philippe B. Laval (KSU) Sets Current Semester 1 / 16 Outline 1 Introduction 2 Definitions, Notations and Examples 3 Special
More informationHowever, this is not always true! For example, this fails if both A and B are closed and unbounded (find an example).
98 CHAPTER 3. PROPERTIES OF CONVEX SETS: A GLIMPSE 3.2 Separation Theorems It seems intuitively rather obvious that if A and B are two nonempty disjoint convex sets in A 2, then there is a line, H, separating
More informationTopology notes. Basic Definitions and Properties.
Topology notes. Basic Definitions and Properties. Intuitively, a topological space consists of a set of points and a collection of special sets called open sets that provide information on how these points
More informationTOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3.
TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3. 301. Definition. Let m be a positive integer, and let X be a set. An m-tuple of elements of X is a function x : {1,..., m} X. We sometimes use x i instead
More informationMA651 Topology. Lecture 4. Topological spaces 2
MA651 Topology. Lecture 4. Topological spaces 2 This text is based on the following books: Linear Algebra and Analysis by Marc Zamansky Topology by James Dugundgji Fundamental concepts of topology by Peter
More informationChapter 1. Preliminaries
Chapter 1 Preliminaries 1.1 Topological spaces 1.1.1 The notion of topological space The topology on a set X is usually defined by specifying its open subsets of X. However, in dealing with topological
More information2.8. Connectedness A topological space X is said to be disconnected if X is the disjoint union of two non-empty open subsets. The space X is said to
2.8. Connectedness A topological space X is said to be disconnected if X is the disjoint union of two non-empty open subsets. The space X is said to be connected if it is not disconnected. A subset of
More informationFinal Exam, F11PE Solutions, Topology, Autumn 2011
Final Exam, F11PE Solutions, Topology, Autumn 2011 Question 1 (i) Given a metric space (X, d), define what it means for a set to be open in the associated metric topology. Solution: A set U X is open if,
More informationReal Analysis, 2nd Edition, G.B.Folland
Real Analysis, 2nd Edition, G.B.Folland Chapter 4 Point Set Topology Yung-Hsiang Huang 4.1 Topological Spaces 1. If card(x) 2, there is a topology on X that is T 0 but not T 1. 2. If X is an infinite set,
More informationLecture 17: Continuous Functions
Lecture 17: Continuous Functions 1 Continuous Functions Let (X, T X ) and (Y, T Y ) be topological spaces. Definition 1.1 (Continuous Function). A function f : X Y is said to be continuous if the inverse
More information2 A topological interlude
2 A topological interlude 2.1 Topological spaces Recall that a topological space is a set X with a topology: a collection T of subsets of X, known as open sets, such that and X are open, and finite intersections
More informationMATH 54 - LECTURE 10
MATH 54 - LECTURE 10 DAN CRYTSER The Universal Mapping Property First we note that each of the projection mappings π i : j X j X i is continuous when i X i is given the product topology (also if the product
More informationTHREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.
THREE LECTURES ON BASIC TOPOLOGY PHILIP FOTH 1. Basic notions. Let X be a set. To make a topological space out of X, one must specify a collection T of subsets of X, which are said to be open subsets of
More informationOn Soft Topological Linear Spaces
Republic of Iraq Ministry of Higher Education and Scientific Research University of AL-Qadisiyah College of Computer Science and Formation Technology Department of Mathematics On Soft Topological Linear
More informationNotes on metric spaces and topology. Math 309: Topics in geometry. Dale Rolfsen. University of British Columbia
Notes on metric spaces and topology Math 309: Topics in geometry Dale Rolfsen University of British Columbia Let X be a set; we ll generally refer to its elements as points. A distance function, or metric
More informationWhat is Set? Set Theory. Notation. Venn Diagram
What is Set? Set Theory Peter Lo Set is any well-defined list, collection, or class of objects. The objects in set can be anything These objects are called the Elements or Members of the set. CS218 Peter
More informationA GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY
A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY KARL L. STRATOS Abstract. The conventional method of describing a graph as a pair (V, E), where V and E repectively denote the sets of vertices and edges,
More informationM3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements.
M3P1/M4P1 (2005) Dr M Ruzhansky Metric and Topological Spaces Summary of the course: definitions, examples, statements. Chapter 1: Metric spaces and convergence. (1.1) Recall the standard distance function
More informationLecture 11 COVERING SPACES
Lecture 11 COVERING SPACES A covering space (or covering) is not a space, but a mapping of spaces (usually manifolds) which, locally, is a homeomorphism, but globally may be quite complicated. The simplest
More informationLectures on Order and Topology
Lectures on Order and Topology Antonino Salibra 17 November 2014 1 Topology: main definitions and notation Definition 1.1 A topological space X is a pair X = ( X, OX) where X is a nonempty set and OX is
More informationLecture IV - Further preliminaries from general topology:
Lecture IV - Further preliminaries from general topology: We now begin with some preliminaries from general topology that is usually not covered or else is often perfunctorily treated in elementary courses
More informationA Little Point Set Topology
A Little Point Set Topology A topological space is a generalization of a metric space that allows one to talk about limits, convergence, continuity and so on without requiring the concept of a distance
More information(i, j)-almost Continuity and (i, j)-weakly Continuity in Fuzzy Bitopological Spaces
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 4, Issue 2, February 2016, PP 89-98 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org (i, j)-almost
More informationLecture 4: examples of topological spaces, coarser and finer topologies, bases and closed sets
Lecture 4: examples of topological spaces, coarser and finer topologies, bases and closed sets Saul Glasman 14 September 2016 Let s give the definition of an open subset of R. Definition 1. Let U R. We
More informationCS 6170: Computational Topology, Spring 2019 Lecture 03
CS 6170: Computational Topology, Spring 2019 Lecture 03 Topological Data Analysis for Data Scientists Dr. Bei Wang School of Computing Scientific Computing and Imaging Institute (SCI) University of Utah
More information2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Fall
c Dr Oksana Shatalov, Fall 2014 1 2. Sets 2.1&2.2: Sets and Subsets. Combining Sets. Set Terminology and Notation DEFINITIONS: Set is well-defined collection of objects. Elements are objects or members
More informationJohns Hopkins Math Tournament Proof Round: Point Set Topology
Johns Hopkins Math Tournament 2019 Proof Round: Point Set Topology February 9, 2019 Problem Points Score 1 3 2 6 3 6 4 6 5 10 6 6 7 8 8 6 9 8 10 8 11 9 12 10 13 14 Total 100 Instructions The exam is worth
More informationFinal Test in MAT 410: Introduction to Topology Answers to the Test Questions
Final Test in MAT 410: Introduction to Topology Answers to the Test Questions Stefan Kohl Question 1: Give the definition of a topological space. (3 credits) A topological space (X, τ) is a pair consisting
More informationChapter 11. Topological Spaces: General Properties
11.1. Open Sets, Closed Sets, Bases, and Subbases 1 Chapter 11. Topological Spaces: General Properties Section 11.1. Open Sets, Closed Sets, Bases, and Subbases Note. In this section, we define a topological
More informationThe Space of Closed Subsets of a Convergent Sequence
The Space of Closed Subsets of a Convergent Sequence by Ashley Reiter and Harold Reiter Many topological spaces are simply sets of points(atoms) endowed with a topology Some spaces, however, have elements
More informationTopology and Topological Spaces
Topology and Topological Spaces Mathematical spaces such as vector spaces, normed vector spaces (Banach spaces), and metric spaces are generalizations of ideas that are familiar in R or in R n. For example,
More informationTopological properties of convex sets
Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory Winter 2018 Topic 5: Topological properties of convex sets 5.1 Interior and closure of convex sets Let
More informationSimplicial Complexes: Second Lecture
Simplicial Complexes: Second Lecture 4 Nov, 2010 1 Overview Today we have two main goals: Prove that every continuous map between triangulable spaces can be approximated by a simplicial map. To do this,
More informationON BINARY TOPOLOGICAL SPACES
Pacific-Asian Journal of Mathematics, Volume 5, No. 2, July-December 2011 ON BINARY TOPOLOGICAL SPACES S. NITHYANANTHA JOTHI & P. THANGAVELU ABSTRACT: Recently the authors introduced the concept of a binary
More informationINTRODUCTION TO TOPOLOGY
INTRODUCTION TO TOPOLOGY MARTINA ROVELLI These notes are an outline of the topics covered in class, and are not substitutive of the lectures, where (most) proofs are provided and examples are discussed
More information2.2 Set Operations. Introduction DEFINITION 1. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, EXAMPLE 2
2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of
More information= [ U 1 \ U 2 = B \ [ B \ B.
5. Mon, Sept. 8 At the end of class on Friday, we introduced the notion of a topology, and I asked you to think about how many possible topologies there are on a 3-element set. The answer is... 29. The
More information4. Definition: topological space, open set, topology, trivial topology, discrete topology.
Topology Summary Note to the reader. If a statement is marked with [Not proved in the lecture], then the statement was stated but not proved in the lecture. Of course, you don t need to know the proof.
More informationLecture 2 - Introduction to Polytopes
Lecture 2 - Introduction to Polytopes Optimization and Approximation - ENS M1 Nicolas Bousquet 1 Reminder of Linear Algebra definitions Let x 1,..., x m be points in R n and λ 1,..., λ m be real numbers.
More informationNotes on Topology. Andrew Forrester January 28, Notation 1. 2 The Big Picture 1
Notes on Topology Andrew Forrester January 28, 2009 Contents 1 Notation 1 2 The Big Picture 1 3 Fundamental Concepts 2 4 Topological Spaces and Topologies 2 4.1 Topological Spaces.........................................
More informationCHAPTER 8. Copyright Cengage Learning. All rights reserved.
CHAPTER 8 RELATIONS Copyright Cengage Learning. All rights reserved. SECTION 8.3 Equivalence Relations Copyright Cengage Learning. All rights reserved. The Relation Induced by a Partition 3 The Relation
More informationT. Background material: Topology
MATH41071/MATH61071 Algebraic topology Autumn Semester 2017 2018 T. Background material: Topology For convenience this is an overview of basic topological ideas which will be used in the course. This material
More information1.1 - Introduction to Sets
1.1 - Introduction to Sets Math 166-502 Blake Boudreaux Department of Mathematics Texas A&M University January 18, 2018 Blake Boudreaux (Texas A&M University) 1.1 - Introduction to Sets January 18, 2018
More information4. Simplicial Complexes and Simplicial Homology
MATH41071/MATH61071 Algebraic topology Autumn Semester 2017 2018 4. Simplicial Complexes and Simplicial Homology Geometric simplicial complexes 4.1 Definition. A finite subset { v 0, v 1,..., v r } R n
More informationEDAA40 At home exercises 1
EDAA40 At home exercises 1 1. Given, with as always the natural numbers starting at 1, let us define the following sets (with iff ): Give the number of elements in these sets as follows: 1. 23 2. 6 3.
More informationRough Connected Topologized. Approximation Spaces
International Journal o Mathematical Analysis Vol. 8 04 no. 53 69-68 HIARI Ltd www.m-hikari.com http://dx.doi.org/0.988/ijma.04.4038 Rough Connected Topologized Approximation Spaces M. J. Iqelan Department
More information2 Review of Set Theory
2 Review of Set Theory Example 2.1. Let Ω = {1, 2, 3, 4, 5, 6} 2.2. Venn diagram is very useful in set theory. It is often used to portray relationships between sets. Many identities can be read out simply
More informationOrientation of manifolds - definition*
Bulletin of the Manifold Atlas - definition (2013) Orientation of manifolds - definition* MATTHIAS KRECK 1. Zero dimensional manifolds For zero dimensional manifolds an orientation is a map from the manifold
More informationTopology Between Two Sets
Journal of Mathematical Sciences & Computer Applications 1 (3): 95 107, 2011 doi: 10.5147/jmsca.2011.0071 Topology Between Two Sets S. Nithyanantha Jothi 1 and P. Thangavelu 2* 1 Department of Mathematics,
More informationBounded subsets of topological vector spaces
Chapter 2 Bounded subsets of topological vector spaces In this chapter we will study the notion of bounded set in any t.v.s. and analyzing some properties which will be useful in the following and especially
More informationIn class 75min: 2:55-4:10 Thu 9/30.
MATH 4530 Topology. In class 75min: 2:55-4:10 Thu 9/30. Prelim I Solutions Problem 1: Consider the following topological spaces: (1) Z as a subspace of R with the finite complement topology (2) [0, π]
More informationSection 13. Basis for a Topology
13. Basis for a Topology 1 Section 13. Basis for a Topology Note. In this section, we consider a basis for a topology on a set which is, in a sense, analogous to the basis for a vector space. Whereas a
More informationPoint-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS
Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS Definition 1.1. Let X be a set and T a subset of the power set P(X) of X. Then T is a topology on X if and only if all of the following
More informationTopology - I. Michael Shulman WOMP 2004
Topology - I Michael Shulman WOMP 2004 1 Topological Spaces There are many different ways to define a topological space; the most common one is as follows: Definition 1.1 A topological space (often just
More informationTopological space - Wikipedia, the free encyclopedia
Page 1 of 6 Topological space From Wikipedia, the free encyclopedia Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity.
More informationFunctions. How is this definition written in symbolic logic notation?
functions 1 Functions Def. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by
More informationand this equivalence extends to the structures of the spaces.
Homeomorphisms. A homeomorphism between two topological spaces (X, T X ) and (Y, T Y ) is a one - one correspondence such that f and f 1 are both continuous. Consequently, for every U T X there is V T
More informationTOWARDS FORMING THE FIELD OF FUZZY CLOSURE WITH REFERENCE TO FUZZY BOUNDARY
TOWARDS FORMING THE FIELD OF FUZZY CLOSURE WITH REFERENCE TO FUZZY BOUNDARY Bhimraj Basumatary Department of Mathematical Sciences, Bodoland University Kokrajhar, BTC, Assam, India, 783370 brbasumatary14@gmail.com
More informationEXTERNAL VISIBILITY. 1. Definitions and notation. The boundary and interior of
PACIFIC JOURNAL OF MATHEMATICS Vol. 64, No. 2, 1976 EXTERNAL VISIBILITY EDWIN BUCHMAN AND F. A. VALENTINE It is possible to see any eleven vertices of an opaque solid regular icosahedron from some appropriate
More informationNotes on Interval Valued Fuzzy RW-Closed, Interval Valued Fuzzy RW-Open Sets in Interval Valued Fuzzy Topological Spaces
International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 23-38 Research India Publications http://www.ripublication.com Notes on Interval Valued Fuzzy RW-Closed,
More informationOn Fuzzy Topological Spaces Involving Boolean Algebraic Structures
Journal of mathematics and computer Science 15 (2015) 252-260 On Fuzzy Topological Spaces Involving Boolean Algebraic Structures P.K. Sharma Post Graduate Department of Mathematics, D.A.V. College, Jalandhar
More informationElementary Topology. Note: This problem list was written primarily by Phil Bowers and John Bryant. It has been edited by a few others along the way.
Elementary Topology Note: This problem list was written primarily by Phil Bowers and John Bryant. It has been edited by a few others along the way. Definition. properties: (i) T and X T, A topology on
More informationFundamental Properties of Graphs
Chapter three In many real-life situations we need to know how robust a graph that represents a certain network is, how edges or vertices can be removed without completely destroying the overall connectivity,
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More information4 Basis, Subbasis, Subspace
4 Basis, Subbasis, Subspace Our main goal in this chapter is to develop some tools that make it easier to construct examples of topological spaces. By Definition 3.12 in order to define a topology on a
More informationChapter 2 Notes on Point Set Topology
Chapter 2 Notes on Point Set Topology Abstract The chapter provides a brief exposition of point set topology. In particular, it aims to make readers from the engineering community feel comfortable with
More informationHomework Set #2 Math 440 Topology Topology by J. Munkres
Homework Set #2 Math 440 Topology Topology by J. Munkres Clayton J. Lungstrum October 26, 2012 Exercise 1. Prove that a topological space X is Hausdorff if and only if the diagonal = {(x, x) : x X} is
More informationA Tour of General Topology Chris Rogers June 29, 2010
A Tour of General Topology Chris Rogers June 29, 2010 1. The laundry list 1.1. Metric and topological spaces, open and closed sets. (1) metric space: open balls N ɛ (x), various metrics e.g. discrete metric,
More informationNOTES ON GENERAL TOPOLOGY
NOTES ON GENERAL TOPOLOGY PETE L. CLARK 1. The notion of a topological space Part of the rigorization of analysis in the 19th century was the realization that notions like convergence of sequences and
More informationThese notes present some properties of chordal graphs, a set of undirected graphs that are important for undirected graphical models.
Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations Peter Bartlett. October 2003. These notes present some properties of chordal graphs, a set of undirected
More informationBases of topologies. 1 Motivation
Bases of topologies 1 Motivation In the previous section we saw some examples of topologies. We described each of them by explicitly specifying all of the open sets in each one. This is not be a feasible
More informationCompactness in Countable Fuzzy Topological Space
Compactness in Countable Fuzzy Topological Space Apu Kumar Saha Assistant Professor, National Institute of Technology, Agartala, Email: apusaha_nita@yahoo.co.in Debasish Bhattacharya Associate Professor,
More informationSlides for Faculty Oxford University Press All rights reserved.
Oxford University Press 2013 Slides for Faculty Assistance Preliminaries Author: Vivek Kulkarni vivek_kulkarni@yahoo.com Outline Following topics are covered in the slides: Basic concepts, namely, symbols,
More informationAlgebra of Sets (Mathematics & Logic A)
Algebra of Sets (Mathematics & Logic A) RWK/MRQ October 28, 2002 Note. These notes are adapted (with thanks) from notes given last year by my colleague Dr Martyn Quick. Please feel free to ask me (not
More informationTopology I Test 1 Solutions October 13, 2008
Topology I Test 1 Solutions October 13, 2008 1. Do FIVE of the following: (a) Give a careful definition of connected. A topological space X is connected if for any two sets A and B such that A B = X, we
More informationIntroduction to Algebraic and Geometric Topology Week 5
Introduction to Algebraic and Geometric Topology Week 5 Domingo Toledo University of Utah Fall 2017 Topology of Metric Spaces I (X, d) metric space. I Recall the definition of Open sets: Definition U
More informationManifolds. Chapter X. 44. Locally Euclidean Spaces
Chapter X Manifolds 44. Locally Euclidean Spaces 44 1. Definition of Locally Euclidean Space Let n be a non-negative integer. A topological space X is called a locally Euclidean space of dimension n if
More informationThis is already grossly inconvenient in present formalisms. Why do we want to make this convenient? GENERAL GOALS
1 THE FORMALIZATION OF MATHEMATICS by Harvey M. Friedman Ohio State University Department of Mathematics friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ May 21, 1997 Can mathematics be
More informationLecture 0: Reivew of some basic material
Lecture 0: Reivew of some basic material September 12, 2018 1 Background material on the homotopy category We begin with the topological category TOP, whose objects are topological spaces and whose morphisms
More informationIntroduction II. Sets. Terminology III. Definition. Definition. Definition. Example
Sets Slides by Christopher M. ourke Instructor: erthe Y. Choueiry Spring 2006 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 1.6 1.7 of Rosen cse235@cse.unl.edu Introduction
More informationCantor s Diagonal Argument for Different Levels of Infinity
JANUARY 2015 1 Cantor s Diagonal Argument for Different Levels of Infinity Michael J. Neely University of Southern California http://www-bcf.usc.edu/ mjneely Abstract These notes develop the classic Cantor
More informationCS522: Advanced Algorithms
Lecture 1 CS5: Advanced Algorithms October 4, 004 Lecturer: Kamal Jain Notes: Chris Re 1.1 Plan for the week Figure 1.1: Plan for the week The underlined tools, weak duality theorem and complimentary slackness,
More information2. Metric and Topological Spaces
2 Metric and Topological Spaces Topology begins where sets are implemented with some cohesive properties enabling one to define continuity Solomon Lefschetz In order to forge a language of continuity,
More informationMetric and metrizable spaces
Metric and metrizable spaces These notes discuss the same topic as Sections 20 and 2 of Munkres book; some notions (Symmetric, -metric, Ψ-spaces...) are not discussed in Munkres book.. Symmetric, -metric,
More informationA WILD CANTOR SET IN THE HILBERT CUBE
PACIFIC JOURNAL OF MATHEMATICS Vol. 24, No. 1, 1968 A WILD CANTOR SET IN THE HILBERT CUBE RAYMOND Y. T. WONG Let E n be the Euclidean w-space. A Cantor set C is a set homeomorphic with the Cantor middle-third
More informationCharacterization of Boolean Topological Logics
Characterization of Boolean Topological Logics Short Form: Boolean Topological Logics Anthony R. Fressola Denison University Granville, OH 43023 University of Illinois Urbana-Champaign, IL USA 61801-61802
More informationNim-Regularity of Graphs
Nim-Regularity of Graphs Nathan Reading School of Mathematics, University of Minnesota Minneapolis, MN 55455 reading@math.umn.edu Submitted: November 24, 1998; Accepted: January 22, 1999 Abstract. Ehrenborg
More informationSTABILITY AND PARADOX IN ALGORITHMIC LOGIC
STABILITY AND PARADOX IN ALGORITHMIC LOGIC WAYNE AITKEN, JEFFREY A. BARRETT Abstract. Algorithmic logic is the logic of basic statements concerning algorithms and the algorithmic rules of deduction between
More informationThe Set-Open topology
Volume 37, 2011 Pages 205 217 http://topology.auburn.edu/tp/ The Set-Open topology by A. V. Osipov Electronically published on August 26, 2010 Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail:
More informationSection 17. Closed Sets and Limit Points
17. Closed Sets and Limit Points 1 Section 17. Closed Sets and Limit Points Note. In this section, we finally define a closed set. We also introduce several traditional topological concepts, such as limit
More information6c Lecture 3 & 4: April 8 & 10, 2014
6c Lecture 3 & 4: April 8 & 10, 2014 3.1 Graphs and trees We begin by recalling some basic definitions from graph theory. Definition 3.1. A (undirected, simple) graph consists of a set of vertices V and
More information