Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008

Size: px
Start display at page:

Download "Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008"

Transcription

1 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 with a distinguished family of subsets. This family of subsets is called a topology for the given set. Definition. Let X be a set. A collection of subsets U = {U} of X is called a topology for X, if it satisfies the following properties: 1. X U and U. 2. The intersection of finitely many members of U is again in U. 3. The union of arbitrarily many members of U is again in U. A topological space is a set together with a topology. The family U is generally called the family of open sets of the topological space. We have also observed that a topology is also determined by specifying the family of closed sets, namely, the family of sets whose complements are open. Proposition 1. Let X be a set with a topology U. Define a family of subsets C = {U c }, where U c denotes X U, the complement of U in X. Then the family C has the following properties. 1. X C and C. 2. The union of finitely many members of C is again in C. 3. The intersection of arbitrarily many members of C is again in C. The family C is generally called the family of closed sets of the topological space X. In some respects it is a matter of convenience whether or not one chooses to work with closed sets or open sets. However, for connectivity, we need to develop a few more properties of closed sets. First we define an operation called closure. Definition. Let X be a topological space and let A X. Define A = C We call A the closure of A. C closed, C A 1

2 Thus, A is the smallest closed set in X containing A, and, if C is closed, C = C. Often, formal properties of the closure operation follow easily from this conceptual defintion. For the sake of calculation of set closures in examples, we characterize the elements of a closed set in terms of open sets. Proposition 2. Let X be a topological space and let A X. A point x X is in A if, and only if, for every open set U with x U, U A. Proof. One way to see this result is to convert it to the contrapositive equivalent. A point x X is not in A if, and only if, there is an open set U, x U, U A =. Now, if x / A, then there is a closed set C, C A, and x / C. Then x U = X C, which is an open set, and, since A C, U A =. On the other hand, if, given x, there is an open set U such that U A =, then C = X U is a closed set, A C, and x / C. So x / A. Here is a characterization of continuous functions in terms of the closure operation that is occasionally useful. In fact, Hatcher uses this on page 35 in his calculation of the fundamental group of high-dimensional spheres. Proposition 3. Let X and Y be topological spaces and let f : X Y be a function. The following conditions are equivalent. 1. The function f is continuous. 2. For any B X, f(b) f(b). Proof. If f is continuous, we know that the preimage of a closed subset of Y is closed. In particular, for any B X, f 1 (f(b)) is closed. Obviously, B f 1 (f(b)). Since B is the smallest closed set containing B, we have B f 1 (f(b)), which is a restatement of condition 2. On the other hand, assume condition 2 holds, and let V Y be open. Let B = X f 1 (V ) = f 1 (Y V ). By condition 2, we have f(b) f(f 1 (Y V )) Y V = Y V, since V is open. Equivalently, B f 1 (Y V ) = B. Since we always have B B, we have, in fact, B = B. That is, B = f 1 (Y V ) = X f 1 (V ) is closed. In turn, f 1 (V ) is open, which means that f is continuous. This characterization of continuity reflects the intuitive idea that a continuous function is one that preserves limits. Here we have that B and its limit points are carried into f(b) and its limit points, if f is continuous. There is a dual notion, called the (set-theoretic) interior of a set, defined as IntA = A= U open, U A and described as the largest open subset of A. We won t use this very often. U 2

3 Topological connectedness Essentially, we start this section by defining what we mean for a subset of a topological space to be disconnected. Roughly speaking, we know how points are separated by open sets in Hausdorff space, so we try to extend the separation idea to subsets. Definition. Let X be a topological space, A X a subset, possibly X itself. A separation of A is a pair U, V of open subsets of X, such that 1. A U V. 2. U V =. 3. neither A U nor A V is empty. Once we know when parts of a subset are separated, then, by definition, we also know when a subset is connected. This may be true in a strict logical sense, but it may take some effort to accept some of the examples. Definition. Let X be a topological space and let A be a subset. The set A is called connected if it has no separation. That is, A is connected, if, for every pair of open subsets U, V such that 1. A U V. 2. U V =. 3. either A U or A V is empty. The empty subset is always connected. Take A = X. If X has a separation X = U V, then U and V are both open and closed in X. On the other hand, if X has no separation, then and X are the only subsets of X that are both open and closed. Sometimes we exploit this fact of connected spaces to prove that a property holds everywhere in a connected topological space by proving that set of points having the property is both open and closed. We will see examples later. The first part of the following proposition says connectedness is a topological property, preserved by continuous functions. The second part is a characterization of connected sets in terms of maps to discrete spaces. Proposition 4. Let f : X Y be a map and let A be a connected subset of X. Then f(a) is a connected subset of Y. Let f : X {0, 1} be a map, where {0, 1} has the discrete topology. If A X is connected, then the map f A is constant. Proposition 5. If A is a connected subset of X and B satisfies A B A, then B is also connected. Proof. Let U, V be open, suppose B U V and U V =. If x B, then x A, and for every open subset W containing x, W A. Now, if x B, either x U or x V. If x U, then U A. But A is connected, and has no separation, so we must have A V =. That is, A X V, which implies B A X V. Thus, B V =. We conclude that B has no separation, and, so, B is connected. 3

4 Proposition 6. Let X be a topological space and let {A i : i I} be a family of connected subsets. If i I A i, then i I A i is connected as well. Proof. Write i I A i = U V, where U and V are open in X. Choose an index i 0. We have A i0 U V, and A i0 is connected. Then all of A i0 is in one or the other of the open sets, so we may assume A i0 U. Now let x i I A i. We have x A i0 U, so x U. Then, for all i I, A i U. Since each A i is connected, we must have A i U and A i V =. Then i I A i U and i I A i V =. In other words, i I A i has no separation, so it s connected. So far we don t know any examples of topologically connected spaces, except for trivial ones. The following proposition cures this. The essential ingredients of the proof are that the real numbers have the least upper bound property and that between any two real numbers there is another. Proposition 7. Let I = [a, b] be an interval in R. Then I is connected. Proof. Let U, V be open subsets in R that provide a separation of I. That is, suppose also I U V and U V =. Without loss of generality suppose a U. Let S = {x I : The interval [a, x] is in U.} The set S is non-empty, because a S, and S is clearly bounded above by b. Therefore, S has a least upper bound c. Now we want to show two things: First, that c S and, second, that c = b. Once we have these facts, I U and and I V =, so our separation was not real. Toward showing these facts, we first show c U. If not, then c V, and we obtain a contradiction, as follows. Since V is open, there is an interval (c ǫ, c+ǫ) V. Then the interval [a, c ǫ] U, and no interval [a, y] with c ǫ y < c can be in U. This contradicts 2 2 that c is the least upper bound for S. We conclude that c U. If c U, which is open, there is an interval (c ǫ, c+ǫ ) U. Since c is the least upper bound of S, there is an x (c ǫ, c) such that x S. That is, the closed interval [a, x] U. Then [a, c] = [a, x] [x, c] U, so that c S. Now it is easy to see that c = b. We have c S, so, if c < b, then there are numbers c < y < b such that y (c ǫ, c+ǫ ) U. Then [a, y] = [a, c] [c, y] U, which contradicts that c is an upper bound for S. Thus, we have our first non-trivial example of a connected space. Examples Proposition 8. The set R is connected. So are open intervals (c, d), half-lines (c, ) and (, c) and rays [c, ) and (, c]. Proof. For each n 1, the interval [ n, n] is connected, by proposition 7. Now n=1 [ n, n] is non-empty, containing the point 0, for example. Therefore, by proposition 6, [ n, n] = R n=1 is connected. The other examples are handled similarly. 4

5 Proposition 9. Let X be a connected space, and let f : X R be a map. If f(x) < 0 and f(y) > 0, then there is z X such that f(z) = 0. Proof. Consider U = f 1 ((, 0)) and V = f 1 ((0, )). We have x U and y V, so neither is empty. Both are open in X, because f is continuous. If X = U V, then X has a separation, contradicting the hypothesis that X is connected. Consequently, there must be z X such that f(z) = 0. Definition. Define a relation on X by x y if and only if there is a connected A X such that x, y A. Proposition 10. This relation is an equivalence relation. Therefore X is partitioned into disjoint subsets, each of which is called a connected component of X. Proof. Proof of transitivity of requires proposition 6, but is otherwise straightforward. A somewhat more intuitive definition of connectedness, based on the idea of mapping connected test spaces into the space under investigation is called path-connectedness Definition. Let X be a topological space. We say X is path-connected, if, for any pair of points x, y X, there is a map f : I X such that f(0) = x and f(1) = y. We call f a path in X from x to y. Alternatively, given a topological space X, define an relation p on X by x p y if there is a path f in X from x to y. The proof the following proposition is an exercise. Proposition 11. For any space X, the relation p is an equivalence relation. The space X is now partitioned into equivalence classes, called the path-components. Here are two examples that show connectedness and path-connectedness are different notions. Example 1. Let I = [0, 1], as usual, and let A = {1, 1/2, 1/3, 1/4,...} I. Define X = I {0} A I {(0, 1)}. X is called the deleted comb space. The space Y = I {0} A I {0} I is called the comb space. Make a sketch showing both Y and X. Clearly Y is connected, by multiple applications of proposition 6. We see that X is connected, because X I {0} A I, and I {0} A I is connected. Now we see that X is not path-connected. To show that X is not path-connected, let f : I X be a path with f(0) = P = (0, 1). Obviously, f 1 (P) is closed in I, since f is continuous. We will also show that f 1 (P) is open in I. Once we have this additional fact we argue as follows. Since I is connected, its only subsets that are both open and closed are the empty set and I itself. Since f 1 (P) is not empty, f 1 (P) = I and the path f starting at P is constant. To see that f 1 (P) is open, let V be an open ball around P that does not touch the x- axis. f 1 (V ) is open in I. Then, for any t f 1 (V ) there is a small interval J = (t ǫ, t+ǫ) such that f(j) V X. Now let (1/n, y) V X. Choose r, 1/(n + 1) < r < 1/n, and 5

6 define H L = (, r) R and H R = (r, ) R. We note that P H L and (1/n, y) H R. Since the ball V does not touch the x-axis, we must have that f(j) H L. This is true because J is connected, its image under f is connected, and the only way to reach H R in X is by taking a path through (r, 0) on the x-axis, which is specifically excluded from V. Thus, for any point (1/n, y) V, ( (1/n, y) P!), (1/n, y) is not in the image f(j). We conclude that f 1 (V ) = f 1 (P) for an open ball V as above. Thus, f 1 (P) is also open. Example 2. Let Γ = {(x, sin(1/x): 0 < x 1} be a part of the graph of the function f(x) = sin(1/x), defined for x 0. Let W = Γ {0} [ 1, 1] R 2. It is easy to verify that W = Γ, so W is connected by proposition 5. An argument similar to the one proving the deleted comb space is not path-connected shows that W is not path-connected. This example is sometimes called the Warsaw sine curve or the Polish sine curve. Product spaces and connectedness We will see that it is straightforward to demonstrate that a finite product of connected spaces is connected. For infinite products, the result is not so obvious. Proposition 12. Let X and Y be connected non-empty topological spaces. Then X Y is also connected. Proof. We make two applications of proposition 6. Choose x 0 X and let y Y. The subsets A y = X {y} and B = {x 0 } Y are connected, since they are homeomorphic to X and Y, respectively. And A y and B have a point in common. So, by proposition 6, C y = A y B is connected. Now consider the family of subsets {C y : y Y }. Each of these is connected, and C y = B is not empty. Therefore, the union C y = X Y is connected. Inductively, any finite product of connected spaces is connected. Proposition 13. Let {X α : α A} be a family of non empty connected spaces indexed by the infinite set A. Then α A X α is connected, where α A X α is given the product topology. Proof. This result requires a few minutes of consideration of the definition of open sets in the product topology. Recall that the basic open sets, whose unions are the general open sets of the product topology, are sets U = α A U α where U α is open in X α and where U α = X α for all but finitely many α. Back to the proof of the proposition, we know that α F X α, is connected for any finite subset F of A. Now, choose a point (x 0 α) α A X α. For each finite subset F of A we have a copy of α F X α sitting in side of the full product α A X α as the set of points (x α ) defined by the equations x α = x 0 α, for α / F. That is, we restrict the coordinates, if the coordinate index is not in F. All subsets constructed this way are connected, by the finite case, and these subsets have the point (x 0 α ) in common. Therefore the union W is connected. However, we are not yet finished. The union W is not all of the full product. Indeed, it consists of all points of the product for which at most finitely many of the coordinates differ from those of the basepoint (x 0 α ). At this point we really need the definition of open sets 6

7 in the product topology. Given the description of the basic open sets containing any point (x α ), it is easy to see that any of these basic open sets has non-empty intersection with W. But this says that, in the product topology the closure W = α A X α, the full product. By proposition 5, α A X α is connected. Local path-connectedness: We have seen by means of the deleted comb space example and the Warsaw sine curve that connectedness and path-connectedness are distinct notions. However, because the interval is connected, every path-connected space is connected. From the properties of equivalence classes in general, it follows that every connected component of a topological space is a union of path components. However, path components are not necessarily open or closed subsets of a connected component. Just identify the path components in the two examples to see this. There is a useful condition that we can impose to make the situation a little better. That is the condition of being locally path connected. Read the definition very carefully; it may not coincide with your first idea. Definition. We say a space X is locally path-connected, if for every point x, and for every open set V containing x, there is a path-connected open set U such that x U V. This says that every point has path-connected neighborhoods that are arbitrarily small, which is more than saying every point has a path-connected neighborhood. This is not the only way to define locally path-connected ; see Hatcher s remarks at the top of page 62. Proposition 14. Let X be a connected space that is locally path-connected. Then X is path-connected. Proof. Let x 0 X be a point, and define U 0 = {x X : There is a path in X from x 0 to x.} We will show that U 0, which is not empty, is both open and closed in X. Since X is connected, U 0 must be all of X. Let x U 0. The set X is an open set containing x, so there is a path-connected open set U, such that x U X. Following a path from x 0 to x by a path from x to any other point y U shows that, if x U 0, then there is an open subset U such that x U U 0. Thus, U 0 is open. Now let y U 0. We show that y U 0. Recall the characterization of points of a closure given in proposition 2. If y U 0, then for any open set V containing y, V U 0. By local path-connectedness, any open set V containing y contains a path-connected open set U, which also contains y. And U U 0. So, if z U U 0, we can connect x 0 to z, and we can connect z to y by a path in U. Joining the paths together shows that y U 0. Thus, U 0 is also closed. As indicated, U 0 is a non-empty subset of X that is both open and closed. Since X is connected, U 0 = X. This argument can be extended to show that if a space is locally path-connected, then the path components coincide with the components. Moreover, each path component is an open subset of the whole space. 7

1 The Local-to-Global Lemma

1 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 information

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).

Connectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ). Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.

More information

CHAPTER 7. Connectedness

CHAPTER 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 information

Topological properties

Topological 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 information

Def. 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 =

Def. 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 information

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology

More information

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain. Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric

More information

NAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key

NAME: 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 information

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

More information

Math General Topology Fall 2012 Homework 8 Solutions

Math General Topology Fall 2012 Homework 8 Solutions Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that

More information

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements

More information

HW Graph Theory SOLUTIONS (hbovik) - Q

HW Graph Theory SOLUTIONS (hbovik) - Q 1, Diestel 3.5: Deduce the k = 2 case of Menger s theorem (3.3.1) from Proposition 3.1.1. Let G be 2-connected, and let A and B be 2-sets. We handle some special cases (thus later in the induction if these

More information

Introductory Analysis I Fall 2014 Homework #5 Solutions

Introductory Analysis I Fall 2014 Homework #5 Solutions Introductory Analysis I Fall 2014 Homework #5 Solutions 6. Let M be a metric space, let C D M. Now we can think of C as a subset of the metric space M or as a subspace of the metric space D (D being a

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 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 information

STRONGLY CONNECTED SPACES

STRONGLY CONNECTED SPACES Undergraduate Research Opportunity Programme in Science STRONGLY CONNECTED SPACES Submitted by Dai Bo Supervised by Dr. Wong Yan-loi Department of Mathematics National University of Singapore Academic

More information

Extension of continuous functions in digital spaces with the Khalimsky topology

Extension of continuous functions in digital spaces with the Khalimsky topology Extension of continuous functions in digital spaces with the Khalimsky topology Erik Melin Uppsala University, Department of Mathematics Box 480, SE-751 06 Uppsala, Sweden melin@math.uu.se http://www.math.uu.se/~melin

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1

MATH 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 information

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit

More information

MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017

MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A

More information

V (v i + W i ) (v i + W i ) is path-connected and hence is connected.

V (v i + W i ) (v i + W i ) is path-connected and hence is connected. Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that

More information

Topology. 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  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 information

Metric Spaces Lecture 17

Metric Spaces Lecture 17 Metric Spaces Lecture 17 Homeomorphisms At the end of last lecture an example was given of a bijective continuous function f such that f 1 is not continuous. For another example, consider the sets T =

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

More information

P-adic Functions - Part 1

P-adic Functions - Part 1 P-adic Functions - Part 1 Nicolae Ciocan 22.11.2011 1 Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant

More information

LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS

LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS ROTHSCHILD CAESARIA COURSE, 2011/2 1. The idea of approximation revisited When discussing the notion of the

More information

Part III. 10 Topological Space Basics. Topological Spaces

Part III. 10 Topological Space Basics. Topological Spaces Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.

More information

Math 5210, Definitions and Theorems on Metric Spaces

Math 5210, Definitions and Theorems on Metric Spaces Math 5210, Definitions and Theorems on Metric Spaces Let (X, d) be a metric space. We will use the following definitions (see Rudin, chap 2, particularly 2.18) 1. Let p X and r R, r > 0, The ball of radius

More information

3 COUNTABILITY AND CONNECTEDNESS AXIOMS

3 COUNTABILITY AND CONNECTEDNESS AXIOMS 3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first

More information

Introduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION

Introduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from

More information

Thus, 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

Thus, 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 information

Filters in Analysis and Topology

Filters 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 information

INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

More information

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1. Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

More information

Math 145. Codimension

Math 145. Codimension Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in

More information

Set, functions and Euclidean space. Seungjin Han

Set, 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 information

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement.

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement. Math 421, Homework #6 Solutions (1) Let E R n Show that (Ē) c = (E c ) o, i.e. the complement of the closure is the interior of the complement. 1 Proof. Before giving the proof we recall characterizations

More information

HW 4 SOLUTIONS. , x + x x 1 ) 2

HW 4 SOLUTIONS. , x + x x 1 ) 2 HW 4 SOLUTIONS The Way of Analysis p. 98: 1.) Suppose that A is open. Show that A minus a finite set is still open. This follows by induction as long as A minus one point x is still open. To see that A

More information

g 2 (x) (1/3)M 1 = (1/3)(2/3)M.

g 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 information

An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees

An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees Francesc Rosselló 1, Gabriel Valiente 2 1 Department of Mathematics and Computer Science, Research Institute

More information

Week 6: Topology & Real Analysis Notes

Week 6: Topology & Real Analysis Notes Week 6: Topology & Real Analysis Notes To this point, we have covered Calculus I, Calculus II, Calculus III, Differential Equations, Linear Algebra, Complex Analysis and Abstract Algebra. These topics

More information

Economics 204 Fall 2012 Problem Set 3 Suggested Solutions

Economics 204 Fall 2012 Problem Set 3 Suggested Solutions Economics 204 Fall 2012 Problem Set 3 Suggested Solutions 1. Give an example of each of the following (and prove that your example indeed works): (a) A complete metric space that is bounded but not compact.

More information

5. Connectedness. Corollary 5.4. Connectedness is a topological property.

5. Connectedness. Corollary 5.4. Connectedness is a topological property. 5. Connectedness We begin our introduction to topology with the study of connectedness traditionally the only topic studied in both analytic and algebraic topology. C. T. C. Wall, 197 The property at the

More information

3. The Sheaf of Regular Functions

3. The Sheaf of Regular Functions 24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

More information

Math 730 Homework 6. Austin Mohr. October 14, 2009

Math 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 information

Cartesian Products and Relations

Cartesian Products and Relations Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) : (a A) and (b B)}. The following points are worth special

More information

Measurable Choice Functions

Measurable Choice Functions (January 19, 2013) Measurable Choice Functions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/choice functions.pdf] This note

More information

Dynamical Systems 2, MA 761

Dynamical Systems 2, MA 761 Dynamical Systems 2, MA 761 Topological Dynamics This material is based upon work supported by the National Science Foundation under Grant No. 9970363 1 Periodic Points 1 The main objects studied in the

More information

CW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.

CW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes. CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological

More information

Measures. Chapter Some prerequisites. 1.2 Introduction

Measures. Chapter Some prerequisites. 1.2 Introduction Lecture notes Course Analysis for PhD students Uppsala University, Spring 2018 Rostyslav Kozhan Chapter 1 Measures 1.1 Some prerequisites I will follow closely the textbook Real analysis: Modern Techniques

More information

Math 3T03 - Topology

Math 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 information

Notes on Complex Analysis

Notes on Complex Analysis Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................

More information

Math General Topology Fall 2012 Homework 1 Solutions

Math General Topology Fall 2012 Homework 1 Solutions Math 535 - General Topology Fall 2012 Homework 1 Solutions Definition. Let V be a (real or complex) vector space. A norm on V is a function : V R satisfying: 1. Positivity: x 0 for all x V and moreover

More information

Homework 5. Solutions

Homework 5. Solutions Homework 5. Solutions 1. Let (X,T) be a topological space and let A,B be subsets of X. Show that the closure of their union is given by A B = A B. Since A B is a closed set that contains A B and A B is

More information

Economics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017

Economics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017 Economics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017 Section 2.9. Connected Sets Definition 1 Two sets A, B in a metric space are separated if Ā B = A B = A set in a metric space is connected

More information

{x : P (x)} P (x) = x is a cat

{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 information

Spring -07 TOPOLOGY III. Conventions

Spring -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 information

A generalization of modal definability

A generalization of modal definability A generalization of modal definability Tin Perkov Polytechnic of Zagreb Abstract. Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

Topology Math Conrad Plaut

Topology Math Conrad Plaut Topology Math 467 2010 Conrad Plaut Contents Chapter 1. Background 1 1. Set Theory 1 2. Finite and Infinite Sets 3 3. Indexed Collections of Sets 4 Chapter 2. Topology of R and Beyond 7 1. The Topology

More information

Topology Exercise Sheet 2 Prof. Dr. Alessandro Sisto Due to March 7

Topology Exercise Sheet 2 Prof. Dr. Alessandro Sisto Due to March 7 Topology Exercise Sheet 2 Prof. Dr. Alessandro Sisto Due to March 7 Question 1: The goal of this exercise is to give some equivalent characterizations for the interior of a set. Let X be a topological

More information

This chapter contains a very bare summary of some basic facts from topology.

This 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 information

arxiv:math/ v1 [math.lo] 5 Mar 2007

arxiv: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

2 Metric Spaces Definitions Exotic Examples... 3

2 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 information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2. 1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of

More information

Lecture 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 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 information

SETS AND FUNCTIONS JOSHUA BALLEW

SETS AND FUNCTIONS JOSHUA BALLEW SETS AND FUNCTIONS JOSHUA BALLEW 1. Sets As a review, we begin by considering a naive look at set theory. For our purposes, we define a set as a collection of objects. Except for certain sets like N, Z,

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;

More information

Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)

Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010) http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

Functional Analysis HW #5

Functional Analysis HW #5 Functional Analysis HW #5 Sangchul Lee October 29, 2015 Contents 1 Solutions........................................ 1 1 Solutions Exercise 3.4. Show that C([0, 1]) is not a Hilbert space, that is, there

More information

18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015)

18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015) 18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015) 1. Introduction The goal for this course is to provide a quick, and hopefully somewhat gentle, introduction to the task of formulating

More information

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov

More information

Math 105A HW 1 Solutions

Math 105A HW 1 Solutions Sect. 1.1.3: # 2, 3 (Page 7-8 Math 105A HW 1 Solutions 2(a ( Statement: Each positive integers has a unique prime factorization. n N: n = 1 or ( R N, p 1,..., p R P such that n = p 1 p R and ( n, R, S

More information

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined

More information

INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS

INDECOMPOSABILITY 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 information

4 Countability axioms

4 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 information

FUNCTIONAL ANALYSIS CHRISTIAN REMLING

FUNCTIONAL ANALYSIS CHRISTIAN REMLING FUNCTIONAL ANALYSIS CHRISTIAN REMLING Contents 1. Metric and topological spaces 2 2. Banach spaces 12 3. Consequences of Baire s Theorem 30 4. Dual spaces and weak topologies 34 5. Hilbert spaces 50 6.

More information

Algebraic Topology Homework 4 Solutions

Algebraic Topology Homework 4 Solutions Algebraic Topology Homework 4 Solutions Here are a few solutions to some of the trickier problems... Recall: Let X be a topological space, A X a subspace of X. Suppose f, g : X X are maps restricting to

More information

Math 320-2: Final Exam Practice Solutions Northwestern University, Winter 2015

Math 320-2: Final Exam Practice Solutions Northwestern University, Winter 2015 Math 30-: Final Exam Practice Solutions Northwestern University, Winter 015 1. Give an example of each of the following. No justification is needed. (a) A closed and bounded subset of C[0, 1] which is

More information

Posets, homomorphisms and homogeneity

Posets, homomorphisms and homogeneity Posets, homomorphisms and homogeneity Peter J. Cameron and D. Lockett School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, U.K. Abstract Jarik Nešetřil suggested

More information

ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM

ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM ALEXANDER KUPERS Abstract. These are notes on space-filling curves, looking at a few examples and proving the Hahn-Mazurkiewicz theorem. This theorem

More information

ON DEVANEY S DEFINITION OF CHAOS AND DENSE PERIODIC POINTS

ON DEVANEY S DEFINITION OF CHAOS AND DENSE PERIODIC POINTS ON DEVANEY S DEFINITION OF CHAOS AND DENSE PERIODIC POINTS SYAHIDA CHE DZUL-KIFLI AND CHRIS GOOD Abstract. We look again at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic

More information

A NEW LINDELOF SPACE WITH POINTS G δ

A 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 information

Topology Homework Assignment 1 Solutions

Topology Homework Assignment 1 Solutions Topology Homework Assignment 1 Solutions 1. Prove that R n with the usual topology satisfies the axioms for a topological space. Let U denote the usual topology on R n. 1(a) R n U because if x R n, then

More information

Math 6510 Homework 11

Math 6510 Homework 11 2.2 Problems 40 Problem. From the long exact sequence of homology groups associted to the short exact sequence of chain complexes n 0 C i (X) C i (X) C i (X; Z n ) 0, deduce immediately that there are

More information

Homework 4: Mayer-Vietoris Sequence and CW complexes

Homework 4: Mayer-Vietoris Sequence and CW complexes Homework 4: Mayer-Vietoris Sequence and CW complexes Due date: Friday, October 4th. 0. Goals and Prerequisites The goal of this homework assignment is to begin using the Mayer-Vietoris sequence and cellular

More information

Sample Problems for the Second Midterm Exam

Sample Problems for the Second Midterm Exam Math 3220 1. Treibergs σιι Sample Problems for the Second Midterm Exam Name: Problems With Solutions September 28. 2007 Questions 1 10 appeared in my Fall 2000 and Fall 2001 Math 3220 exams. (1) Let E

More information

Knowledge spaces from a topological point of view

Knowledge spaces from a topological point of view Knowledge spaces from a topological point of view V.I.Danilov Central Economics and Mathematics Institute of RAS Abstract In this paper we consider the operations of restriction, extension and gluing of

More information

Math 535: Topology Homework 1. Mueen Nawaz

Math 535: Topology Homework 1. Mueen Nawaz Math 535: Topology Homework 1 Mueen Nawaz Mueen Nawaz Math 535 Topology Homework 1 Problem 1 Problem 1 Find all topologies on the set X = {0, 1, 2}. In the list below, a, b, c X and it is assumed that

More information

On minimal models of the Region Connection Calculus

On 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 information

Cutting and pasting. 2 in R. 3 which are not even topologically

Cutting and pasting. 2 in R. 3 which are not even topologically Cutting and pasting We begin by quoting the following description appearing on page 55 of C. T. C. Wall s 1960 1961 Differential Topology notes, which available are online at http://www.maths.ed.ac.uk/~aar/surgery/wall.pdf.

More information

Topology 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 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 information

A n = A N = [ N, N] A n = A 1 = [ 1, 1]. n=1

A n = A N = [ N, N] A n = A 1 = [ 1, 1]. n=1 Math 235: Assignment 1 Solutions 1.1: For n N not zero, let A n = [ n, n] (The closed interval in R containing all real numbers x satisfying n x n). It is easy to see that we have the chain of inclusion

More information

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

More information

Section 2: Classes of Sets

Section 2: Classes of Sets Section 2: Classes of Sets Notation: If A, B are subsets of X, then A \ B denotes the set difference, A \ B = {x A : x B}. A B denotes the symmetric difference. A B = (A \ B) (B \ A) = (A B) \ (A B). Remarks

More information

01. Review of metric spaces and point-set topology. 1. Euclidean spaces

01. Review of metric spaces and point-set topology. 1. Euclidean spaces (October 3, 017) 01. Review of metric spaces and point-set topology Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 017-18/01

More information