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

Size: px
Start display at page:

Download "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"

Transcription

1 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 X is said to be a connected subset if it is connected in the relative topology. It is clear that X is disconnected iff X has a closed open subset U such that U and U X. In this case, U, the boundary of U, equals U \ int(u) = U \ U =. We say a topological space (X, T ) is the disjoint union of two non-empty topological spaces (X 1, T 1 ) and (X 2, T 2 ) if X is the

2 disjoint union of X 1 and X 2 as sets and if T = {U 1 U 2 : U 1 T 1, U 2 T 2 }. In this case, X is disconnected, since X 1 and X 2 are two non-empty open sets. Conversely, suppose that X is disconnected. Then there are two disjoint non-empty open sets U and V such that X = U V. A subset W of X is open iff W U and W V are open. Hence X is the disjoint union of two non-empty topological spaces U and V with relative topologies. Therefore, X is disconnected iff X is expressible as the disjoint union of two non-empty topological spaces. Example. (a) R is connected: if U is an open set such that U R and U, then U, since U is a disjoint union of

3 open intervals. (b) R n is connected. If R n were the disjoint union of two non-empty open sets U and V, then there would be p U and q V, and the line X passing through p, q, which is homeomorphic to R, would be the disjoint union of two non-empty relatively open sets U X and V X. (c) A discrete space with more than one point is disconnected. (d) An uncountable space with cocountable topology is connected, since the intersection of any two non-empty open subsets is cocountable, hence non-empty. (e) An infinite space with cofinite topology is connected, since the intersection of any two non-empty open subsets is cofinite,

4 hence non-empty Theorem. Let f be a continuous function from a connected topological space X to a topological space Y. Then f (X ) is connected Theorem. Let {E α } be a family of connected subsets of a topological space X such that E α E β for each pair α, β of indices. Then E α is connected. Let X be a topological space and let x X. The connected component of x in X, denoted by C(x), is the union of all connected subsets of X that contain x. By Theorem 8.2, C(x) is connected. It is evidently the largest connected subset of X containing x.

5 If E is a connected subset of X that meets C(x), then E C(x) is connected, so that it must be included in C(x). Hence C(x) includes each connected subset of X it meets. If C(x) meets C(y), then C(x) C(y) and C(y) C(x), and hence C(x) = C(y) Theorem. Two connected components of X either coincide or are disjoint. The connected components of X form a partition of X into maximal connected subsets. 8.3a. Theorem. Each connected component of a topological space is closed. A topological space is said to be totally disconnected if the connected components are all singletons.

6 Example. Any countable metric space is totally disconnected. Hence R 0, with inherited topology from R, is totally disconnected. Proof. Let X be a countable metric space with metric d. Fix x, y X, x y. Let a = d(x, y). Since the distances from x to other points form a countable set, there is a b in the interval (0, a) such that d(x, z) b for all z X. The sets {z : d(x, z) < b} and {z : d(x, z) > b} are a pair of complementary open sets separating x and y, hence y C(x). Therefore, C(x) = {x} Theorem. Any interval in R is connected. Proof. Each open interval is connected because it is

7 Then X is connected, but not locally connected. homeomorphic to R. Suppose that I is a non-open interval and a is an end point of I. Let J = int(i ). Then J is an open interval and a belongs to the closure of J. Let x J. Then J C(x), since J is connected. Since C(x) is closed, we see that a J C(x). It follows that C(x) = I and I is connected. A topological space X is said to be locally connected if, for each p X and each open neighborhood U of p, there is a connected open neighborhood V of p with V U. Example. Define a subset X of R 2 by X = {(x 1, 0) : x 1 R} {(x 1, x 2 ) : 0 x 2 1, x 1 rational}.

8 8.5. Theorem. Each connected component of a locally connected topological space is open.

Real Analysis, 2nd Edition, G.B.Folland

Real 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 information

Topology 550A Homework 3, Week 3 (Corrections: February 22, 2012)

Topology 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 information

THREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.

THREE 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 information

A Tour of General Topology Chris Rogers June 29, 2010

A 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 information

Lectures on Order and Topology

Lectures 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 information

Homework Set #2 Math 440 Topology Topology by J. Munkres

Homework 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 information

Chapter 1. Preliminaries

Chapter 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 information

TOPOLOGY, DR. BLOCK, FALL 2015, NOTES, PART 3.

TOPOLOGY, 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 information

M3P1/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. 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 information

Introduction to Algebraic and Geometric Topology Week 5

Introduction 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 information

Topology Homework 3. Section Section 3.3. Samuel Otten

Topology 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 information

Notes 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 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 information

Topology notes. Basic Definitions and Properties.

Topology 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 information

Lecture 15: The subspace topology, Closed sets

Lecture 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 information

Final 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 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 information

Point-Set Topology II

Point-Set Topology II Point-Set Topology II Charles Staats September 14, 2010 1 More on Quotients Universal Property of Quotients. Let X be a topological space with equivalence relation. Suppose that f : X Y is continuous and

More information

Lecture 17: Continuous Functions

Lecture 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 information

INTRODUCTION TO TOPOLOGY

INTRODUCTION 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 information

9.5 Equivalence Relations

9.5 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 information

A Little Point Set Topology

A 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

Generell Topologi. Richard Williamson. May 6, 2013

Generell Topologi. Richard Williamson. May 6, 2013 Generell Topologi Richard Williamson May 6, 2013 1 8 Thursday 7th February 8.1 Using connectedness to distinguish between topological spaces I Proposition 8.1. Let (, O ) and (Y, O Y ) be topological spaces.

More information

4. Definition: topological space, open set, topology, trivial topology, discrete topology.

4. 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 information

Chapter 11. Topological Spaces: General Properties

Chapter 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 information

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.

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. 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 information

Point-Set Topology 1. TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS

Point-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 information

Metric and metrizable spaces

Metric 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 information

Math 205B - Topology. Dr. Baez. January 19, Christopher Walker. p(x) = (cos(2πx), sin(2πx))

Math 205B - Topology. Dr. Baez. January 19, Christopher Walker. p(x) = (cos(2πx), sin(2πx)) Math 205B - Topology Dr. Baez January 19, 2007 Christopher Walker Theorem 53.1. The map p : R S 1 given by the equation is a covering map p(x) = (cos(2πx), sin(2πx)) Proof. First p is continuous since

More information

In class 75min: 2:55-4:10 Thu 9/30.

In 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 information

Topology - I. Michael Shulman WOMP 2004

Topology - 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 information

Math 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) 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 information

Johns Hopkins Math Tournament Proof Round: Point Set Topology

Johns 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 information

Suppose we have a function p : X Y from a topological space X onto a set Y. we want to give a topology on Y so that p becomes a continuous map.

Suppose we have a function p : X Y from a topological space X onto a set Y. we want to give a topology on Y so that p becomes a continuous map. V.3 Quotient Space Suppose we have a function p : X Y from a topological space X onto a set Y. we want to give a topology on Y so that p becomes a continuous map. Remark If we assign the indiscrete topology

More information

INTRODUCTION Joymon Joseph P. Neighbours in the lattice of topologies Thesis. Department of Mathematics, University of Calicut, 2003

INTRODUCTION Joymon Joseph P. Neighbours in the lattice of topologies Thesis. Department of Mathematics, University of Calicut, 2003 INTRODUCTION Joymon Joseph P. Neighbours in the lattice of topologies Thesis. Department of Mathematics, University of Calicut, 2003 INTRODUCTION The collection C(X) of all topologies on a fixed non-empty

More information

Notes on Topology. Andrew Forrester January 28, Notation 1. 2 The Big Picture 1

Notes 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 information

Lecture : Topological Space

Lecture : Topological Space Example of Lecture : Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Example of 1 2 3 Example of 4 5 6 Example of I Topological spaces and continuous

More information

Topology problem set Integration workshop 2010

Topology problem set Integration workshop 2010 Topology problem set Integration workshop 2010 July 28, 2010 1 Topological spaces and Continuous functions 1.1 If T 1 and T 2 are two topologies on X, show that (X, T 1 T 2 ) is also a topological space.

More information

Topology I Test 1 Solutions October 13, 2008

Topology 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 information

T. Background material: Topology

T. 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 information

Introduction to Rational Billiards II. Talk by John Smillie. August 21, 2007

Introduction to Rational Billiards II. Talk by John Smillie. August 21, 2007 Introduction to Rational Billiards II Talk by John Smillie August 21, 2007 Translation surfaces and their singularities Last time we described the Zemlyakov-Katok construction for billiards on a triangular

More information

2 Review of Set Theory

2 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 information

Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary?

Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Don t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case?

More information

Open and Closed Sets

Open 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 information

Chapter One Topological Spaces

Chapter One Topological Spaces Chapter One Topological Spaces Definition. A collection T of subsets of a set X is a topology if a) and X are elements of T; b) the intersection of any two elements of T is an element of T; and c) the

More information

Manifolds. Chapter X. 44. Locally Euclidean Spaces

Manifolds. 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 information

A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY

A 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 information

On Soft Topological Linear Spaces

On 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 information

Final Exam, F11PE Solutions, Topology, Autumn 2011

Final 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 information

NOTES ON GENERAL TOPOLOGY

NOTES 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 information

Chapter 2 Topological Spaces and Continuity

Chapter 2 Topological Spaces and Continuity Chapter 2 Topological Spaces and Continuity Starting from metric spaces as they are familiar from elementary calculus, one observes that many properties of metric spaces like the notions of continuity

More information

MA651 Topology. Lecture 4. Topological spaces 2

MA651 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 information

Section 16. The Subspace Topology

Section 16. The Subspace Topology 16. The Subspace Product Topology 1 Section 16. The Subspace Topology Note. Recall from Analysis 1 that a set of real numbers U is open relative to set X if there is an open set of real numbers O such

More information

Lecture 11 COVERING SPACES

Lecture 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 information

2. Metric and Topological Spaces

2. 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 information

Section 17. Closed Sets and Limit Points

Section 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 information

MATH 54 - LECTURE 4 DAN CRYTSER

MATH 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 information

CONNECTIVE SPACES JOSEPH MUSCAT AND DAVID BUHAGIAR. Communicated by Takuo Miwa (Received: November 7, 2005)

CONNECTIVE SPACES JOSEPH MUSCAT AND DAVID BUHAGIAR. Communicated by Takuo Miwa (Received: November 7, 2005) Mem. Fac. Sci. Eng. Shimane Univ. Series B: Mathematical Science 39 (2006), pp. 1 13 CONNECTIVE SPACES JOSEPH MUSCAT AND DAVID BUHAGIAR Communicated by Takuo Miwa (Received: November 7, 2005) Abstract.

More information

Bounded subsets of topological vector spaces

Bounded 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 information

The Set-Open topology

The 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 information

Section 13. Basis for a Topology

Section 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 information

Logic and Discrete Mathematics. Section 2.5 Equivalence relations and partitions

Logic and Discrete Mathematics. Section 2.5 Equivalence relations and partitions Logic and Discrete Mathematics Section 2.5 Equivalence relations and partitions Slides version: January 2015 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence

More information

and this equivalence extends to the structures of the spaces.

and 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 information

Topological properties of convex sets

Topological 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 information

Excerpts from. Introduction to Modern Topology and Geometry. Anatole Katok Alexey Sossinsky

Excerpts from. Introduction to Modern Topology and Geometry. Anatole Katok Alexey Sossinsky Excerpts from Introduction to Modern Topology and Geometry Anatole Katok Alexey Sossinsky Contents Chapter 1. BASIC TOPOLOGY 3 1.1. Topological spaces 3 1.2. Continuous maps and homeomorphisms 6 1.3.

More information

simply ordered sets. We ll state only the result here, since the proof is given in Munkres.

simply ordered sets. We ll state only the result here, since the proof is given in Munkres. p. 1 Math 490 Notes 20 More About Compactness Recall that in Munkres it is proved that a simply (totally) ordered set X with the order topology is connected iff it satisfies: (1) Every subset bounded above

More information

= [ U 1 \ U 2 = B \ [ B \ B.

= [ 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 information

Math 5801 General Topology and Knot Theory

Math 5801 General Topology and Knot Theory Lecture 23-10/17/2012 Math 5801 Ohio State University October 17, 2012 Course Info Reading for Friday, October 19 Chapter 3.26, pgs. 163-170 HW 8 for Monday, October 22 Chapter 2.24: 3, 5a-d, 8a-d, 12a-f

More information

Topology and Topological Spaces

Topology 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 information

2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Fall

2. 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 information

Topological space - Wikipedia, the free encyclopedia

Topological 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 information

Comparing sizes of sets

Comparing sizes of sets Comparing sizes of sets Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A

More information

Review 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. 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 information

ISSN X (print) COMPACTNESS OF S(n)-CLOSED SPACES

ISSN X (print) COMPACTNESS OF S(n)-CLOSED SPACES Matematiqki Bilten ISSN 0351-336X (print) 41(LXVII) No. 2 ISSN 1857-9914 (online) 2017(30-38) UDC: 515.122.2 Skopje, Makedonija COMPACTNESS OF S(n)-CLOSED SPACES IVAN LONČAR Abstract. The aim of this paper

More information

Number Theory and Graph Theory

Number Theory and Graph Theory 1 Number Theory and Graph Theory Chapter 6 Basic concepts and definitions of graph theory By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com

More information

Sets. De Morgan s laws. Mappings. Definition. Definition

Sets. De Morgan s laws. Mappings. Definition. Definition Sets Let X and Y be two sets. Then the set A set is a collection of elements. Two sets are equal if they contain exactly the same elements. A is a subset of B (A B) if all the elements of A also belong

More information

Manifolds (Relates to text Sec. 36)

Manifolds (Relates to text Sec. 36) 22M:132 Fall 07 J. Simon Manifolds (Relates to text Sec. 36) Introduction. Manifolds are one of the most important classes of topological spaces (the other is function spaces). Much of your work in subsequent

More information

Reconstruction of Filament Structure

Reconstruction of Filament Structure Reconstruction of Filament Structure Ruqi HUANG INRIA-Geometrica Joint work with Frédéric CHAZAL and Jian SUN 27/10/2014 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches

More information

Dual trees must share their ends

Dual trees must share their ends 1 Dual trees must share their ends Reinhard Diestel Julian Pott Abstract We extend to infinite graphs the matroidal characterization of finite graph duality, that two graphs are dual i they have complementary

More information

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.

Topology 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 information

1 Matchings in Graphs

1 Matchings in Graphs Matchings in Graphs J J 2 J 3 J 4 J 5 J J J 6 8 7 C C 2 C 3 C 4 C 5 C C 7 C 8 6 J J 2 J 3 J 4 J 5 J J J 6 8 7 C C 2 C 3 C 4 C 5 C C 7 C 8 6 Definition Two edges are called independent if they are not adjacent

More information

CSC Discrete Math I, Spring Sets

CSC Discrete Math I, Spring Sets CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its

More information

1.7 The Heine-Borel Covering Theorem; open sets, compact sets

1.7 The Heine-Borel Covering Theorem; open sets, compact sets 1.7 The Heine-Borel Covering Theorem; open sets, compact sets This section gives another application of the interval halving method, this time to a particularly famous theorem of analysis, the Heine Borel

More information

Epimorphisms in the Category of Hausdorff Fuzzy Topological Spaces

Epimorphisms in the Category of Hausdorff Fuzzy Topological Spaces Annals of Pure and Applied Mathematics Vol. 7, No. 1, 2014, 35-40 ISSN: 2279-087X (P), 2279-0888(online) Published on 9 September 2014 www.researchmathsci.org Annals of Epimorphisms in the Category of

More information

However, this is not always true! For example, this fails if both A and B are closed and unbounded (find an example).

However, 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 information

Graph Connectivity G G G

Graph Connectivity G G G Graph Connectivity 1 Introduction We have seen that trees are minimally connected graphs, i.e., deleting any edge of the tree gives us a disconnected graph. What makes trees so susceptible to edge deletions?

More information

Some new higher separation axioms via sets having non-empty interior

Some new higher separation axioms via sets having non-empty interior Bhat & Das, Cogent Mathematics (015), : 109695 http://dx.doi.org/10.1080/3311835.015.109695 PURE MATHEMATICS RESEARCH ARTICLE Some new higher separation axioms via sets having non-empty interior Pratibha

More information

Lecture 5: Simplicial Complex

Lecture 5: Simplicial Complex Lecture 5: Simplicial Complex 2-Manifolds, Simplex and Simplicial Complex Scribed by: Lei Wang First part of this lecture finishes 2-Manifolds. Rest part of this lecture talks about simplicial complex.

More information

Part 3. Topological Manifolds

Part 3. Topological Manifolds Part 3 Topological Manifolds This part is devoted to study of the most important topological spaces, the spaces which provide a scene for most of geometric branches in mathematics such as Differential

More information

4 Basis, Subbasis, Subspace

4 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 information

CONNECTED SPACES AND HOW TO USE THEM

CONNECTED SPACES AND HOW TO USE THEM CONNECTED SPACES AND HOW TO USE THEM 1. How to prove X is connected Checking that a space X is NOT connected is typically easy: you just have to find two disjoint, non-empty subsets A and B in X, such

More information

Cantor s Diagonal Argument for Different Levels of Infinity

Cantor 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 information

K 4 C 5. Figure 4.5: Some well known family of graphs

K 4 C 5. Figure 4.5: Some well known family of graphs 08 CHAPTER. TOPICS IN CLASSICAL GRAPH THEORY K, K K K, K K, K K, K C C C C 6 6 P P P P P. Graph Operations Figure.: Some well known family of graphs A graph Y = (V,E ) is said to be a subgraph of a graph

More information

1.1 Topological spaces. Open and closed sets. Bases. Closure and interior of a set

1.1 Topological spaces. Open and closed sets. Bases. Closure and interior of a set December 14, 2012 R. Engelking: General Topology I started to make these notes from [E1] and only later the newer edition [E2] got into my hands. I don t think that there were too much changes in numbering

More information

Lecture-12: Closed Sets

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 information

TOPOLOGY CHECKLIST - SPRING 2010

TOPOLOGY CHECKLIST - SPRING 2010 TOPOLOGY CHECKLIST - SPRING 2010 The list below serves as an indication of what we have covered in our course on topology. (It was written in a hurry, so there is a high risk of some mistake being made

More information

New Classes of Closed Sets tgr-closed Sets and t gr-closed Sets

New Classes of Closed Sets tgr-closed Sets and t gr-closed Sets International Mathematical Forum, Vol. 10, 2015, no. 5, 211-220 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2015.5212 New Classes of Closed Sets tgr-closed Sets and t gr-closed Sets Ahmed

More information

Slides for Faculty Oxford University Press All rights reserved.

Slides 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 information

CS 6170: Computational Topology, Spring 2019 Lecture 03

CS 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 information

SET DEFINITION 1 elements members

SET DEFINITION 1 elements members SETS SET DEFINITION 1 Unordered collection of objects, called elements or members of the set. Said to contain its elements. We write a A to denote that a is an element of the set A. The notation a A denotes

More information

I-CONTINUITY IN TOPOLOGICAL SPACES. Martin Sleziak

I-CONTINUITY IN TOPOLOGICAL SPACES. Martin Sleziak I-CONTINUITY IN TOPOLOGICAL SPACES Martin Sleziak Abstract. In this paper we generalize the notion of I-continuity, which was defined in [1] for real functions, to maps on topological spaces. We study

More information

Classifications in Low Dimensions

Classifications in Low Dimensions Chapter XI Classifications in Low Dimensions In different geometric subjects there are different ideas which dimensions are low and which high. In topology of manifolds low dimension means at most 4. However,

More information