STRONGLY CONNECTED SPACES

Size: px
Start display at page:

Download "STRONGLY CONNECTED SPACES"

Transcription

1 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 Year 1999/2000

2 ABSTRACT When discussing the concept of connectedness, we often come across the equivalent criterion that a space is connected if and only if any continuous map from it to the discrete space {0,1} is constant. It would be interesting to see what concept arises if the discrete space of two points is replaced by some other spaces. Let Z be a T 1 space which has more than one point, then a space X is said to be Z-connected if and only if any continuous map from X to Z is constant. It can be shown that this idea generates some stronger notion of connectedness and this stronger notion has many similarities with the usual concept of connectedness. The first nontrivial example of Z-connected space can be constructed by taking Z to be the space Ζ of integers equipped with the complement finite topology. Define a space X to be strongly connected if and only if it is Ζ connected. We shall show in this paper that strongly connected spaces have many interesting properties.

3 TABLE OF CONTENTS CHAPTER 1: INTRODUCTION REVIEW OF THE REPORT ACKNOWLEDGEMENT... 2 CHAPTER 2: REVIEW OF CONNECTED SPACES CONCEPT OF CONNECTEDNESS PROPERTIES OF CONNECTED SPACES EXAMPLES OF CONNECTED SPACES. 6 CHAPTER 3: Z-CONNECTED SPACES CONCEPT OF Z-CONNECTEDNESS EXAMPLES OF Z-CONNECTED SPACES PROPERTIES OF Z-CONNECTEDNESS. 13 CHAPTER 4: STRONGLY CONNECTED SPACES CONCEPT OF STRONGLY CONNECTEDNESS USUAL PROPERTY FOR STRONGLY CONNECTED SPACES STRONGLY CONNECTEDNESS IN COMPACT SPACES STRONGLY CONNECTEDNESS IN COMPLETE METRIC SPACES 23

4 4.5 EXAMPLES OF STRONGLY CONNECTED SPACES..24 CHAPTTER 5: CONCLUSION SUMMARY OF THE WORK FURTHER DISCUSSION REFERENCES... 31

5

6 CHAPTER INTRODUCTION In the study of calculus, there is a basic theorem about continuous functions called Intermediate Value Theorem. If f: [a, b] R is continuous and r is a real number between f(a) and f(b), then there exists an element c [a, b] such that f(c) = r. This theorem is used in a number of places, for instance when constructing inverse functions such as sin -1 (x). The property of the space [a, b] on which the Intermediate Value Theorem depends is the connectedness, and as the Intermediate Value Theorem is a fundamental theorem for analysis, the notion of connectedness is very important in higher analysis, geometry and topology -- indeed, in almost any subject for which the notion of topological space is relevant. However, some connected spaces are not intuitively connected. To describe such cases, we need a stronger notion of connectedness, which is exactly what this project is about. 1.1 OVERVIEW OF THE REPORT The main body of this report is divided into three parts: 1. Review of connected spaces 2. Z-connected spaces 3. Strongly connected spaces The first part offers a quick review of connectedness, including the definition of connectedness, some properties and examples of connected spaces. This is the starting point of the whole project. The second part introduces the concept of Z-connectedness. It shows that how the definition is obtained, what kind of spaces that the Z-connectedness gives us, and how it 1

7 is similar to the connectedness. It will also be shown in this part that the Z-connectedness is a stronger notion of connectedness. In the third part, we will introduce the other new concept called strongly connectedness. A strongly connected space is actually a special case of Z-connected space, which is constructed by choosing a suitable space Z. In this part, the definition of strongly connected spaces is compared with that of connected spaces. Also, some interesting properties of indicating a strongly connected space will be given. 1.2 ACKNOWLEDGEMENT I would like to express my deepest gratitude to my supervisor, Dr Wong Yan-loi, for his utmost patience and guidance in my project. It is him who lets me know about topology from the very basis and gives me great help throughout the year. This course is indeed a bit difficult, but with his help, I find it invaluable. I would also like to thank my classmates for their encouragement and advice. 2

8 CHAPTER REVIEW OF CONNECTED SPACES CONCEPT OF CONNECTEDNESS Intuitively, a connected space is the one that cannot be separated into two or more parts. It is easy to see that the subspace [0, 1] in the real line is connected, whereas the subspace [0, 1] [2, 3] is not connected. However, a formal definition is necessary when dealing with problems such as whether the subspace Q of R is connected, or whether the space l with the lower limit topology is connected. DEFINITION 2.1 A space X is connected if and only if any continuous map f from X to the discrete space {0, 1} is constant. A subset A in a space X is said to be connected if A with the subspace topology is a connected space. There is another equivalent criterion, which is often taken as the definition. It makes use of the concept of separation. A separation of a space X means a pair of nonempty open sets U and V, such that U V = X, and U V =, In symbols, X = U V. DEFINITION 2.2 A topological space X is connected if and only if it has no separation. Proof. Suppose X is connected and it has a separation U V. Define f : X {0, 1} by f (x) = 0 if x U, f (x) = 1 if x V. 3

9 Then this f is continuous and not constant. It contradicts the connectedness of X. Conversely, if X is not connected, then there exists a continuous map f : X {0,1}, and f is not constant. f -1 (0) and f -1 (1) is nonempty and f -1 (0) f -1 (1) = X, f -1 (0) f -1 (1) = Thus X has a separation f -1 (0) f -1 (1). 2.2 PROPERTIES OF CONNECTED SPACES Connectedness is a very useful concept, and it has many important properties. These properties, as well as the proofs, can be found in most books about general topology, thus only the outline of the proofs are shown in this chapter. PROPOSITION 2.3: A continuous image of a connected space is connected. Proof: Let f : X Y be a continuous surjective map and X is connected. Suppose Y is not connected, then by definition, Y has a separation, say U V. Let M = ƒ -1 (U), N = ƒ -1 (V), then U, V are disjoint open sets of X, and form a separation of X. This contradicts the fact that X is connected. Therefore, Y must be connected. PROPOSITION 2.4: The union of any family of connected sets with a common point is connected. Proof: Let { X α } be a family of connected set and p X α for all α. Let f : X α { 0, 1 } be any continuous map and f α : X α { 0, 1 } be the restriction of f to X α. Since f is continuous, each f α is continuous. X α is connected, so f α is constant. Now p X α for all α and f α (X α ) = f(p) for all α, and f( X α ) = f(p), i.e. f is constant. Therefore X α is connected. PROPOSITION 2.5: Let A and B be subsets in a space X such that A B A, where A is the closure of A. If A is connected, then B is connected. Proof: If B is not connected, it has a separation U V. A is connected, A U or A V. 4

10 Without loss of generality, let A U. As A U B, we take closure of A and U in B, A B U. Also, A B = A B = B U, we have B = U. Thus U V is not a separation, and B is connected. For each point p in a space X, the component C(p) of X is the largest connected set in X which contains the point p. PROPOSITION 2.6: For each point p in a space X, the component C(p) of X is a closed set of X. Proof: By definition, C(p) is the largest connected set of X containing the point p. By the last proposition, the closure of C(p) is connected. Hence C( p ) = C(p) and C(p) is closed. PROPOSITION 2.7: The topological product of an arbitrary family of connected spaces is connected. Proof: A proof of the above proposition can be found in [6]. The concept of locally connectedness is often mentioned when talking about connectedness. A locally connected space is defined in term of neighborhood. DEFINITION 2.8 A space X is locally connected at a point p if and only if every neighborhood of p contains a connected neighborhood of p. X is said to be locally connected if it is locally connected at each of its points. The following two propositions are very useful when a locally connected space is involved. PROPOSITION 2.9: Every open subspace of a locally connected space is locally connected. Proof: This is an immediate consequence of the definition. 5

11 PROPOSITION 2.10: For any space X, the following statements are equivalent: (1) X is locally connected. (2) The components of every open subspace of X are open. (3) The connected open sets of X form a basis of the topology of X. Proof: (1) (2). Let X be a locally connected space and let U be an open subspace of X. By Propostion 2.9, U is a locally connected space. Also, the components of U are open sets of U. Since U is open, these components are also open in X. (2) (3). Assume the components of every open subspace of X are open and let U be any open set of X. Since the components of U are connected, U is the union of a collection of connected open sets of X. This proves (3). (3) (1). Assume (3) holds and let U be any open neighborhood of an arbitrary point p in X. By (3), U is the union of a collection of connected open sets. Hence there exists a connected open set V such that p V U. Thus X is locally connected at p. Since p is arbitrary, X is locally connected. Recall in the two or three-dimensional Euclidean space, any two points are connected if they can be joined by a continuous curve. A path connected space is defined similarly. DEFINITION 2.11: A space X is path connected if and only if for any two points a and b in X, there exists a continuous path α : I X such that α(0) = a, α(1) = b. A path connected space have the following property. PROPOSITION 2.12: Every path connected space is connected. 2.3 EXAMPLES OF CONNECTED SPACES With the above definitions and propositions, some examples of connected spaces can be easily found.u (1) Any indiscrete space is connected. 6

12 (2) Any interval of R is connected. (3) The unit n-cube I n of the Euclidean space R n is connected. 1 (4) Let G be the graph of y = sin for x > 0, and I be the line segment joining the points x (0, -1) and (0, 1). Then the space G I is connected. The graph is shown as follows. Fig 1. graph for the example. There are also some spaces that are not connected, for example, (5) The subspace Q of R is not connected. (6) l is not connected. PROOF OF THE EXAMPLES: (1) An indiscrete space does not have any proper open sets, so it cannot have a separation. Therefore an indiscrete space is connected by definition. 7

13 (2) Any interval is a path connected subspace of the real line, thus it is connected by proposition (3) This is the consequence of proposition 2.7 and the last example. (4) The graph G is path connected and connected. I is the set of limit points of G since for any point p on I and any neighborhood U of p, U always intersects G. By proposition 2.5, G I is connected. (5) An irrational number can always be found to separate the set into two parts and thus form a separation. (6) Any open set in l, say [a, b), has a separation [a, c) [c, b). 8

14 CHAPTER Z-CONNECTED SPACES CONCEPT OF Z-CONNECTEDNESS Recall the definition of connectedness, a space X is connected if and only if any continuous map from X to the two-point space with the discrete topology is constant. The concept of Z-connectedness is obtained by replacing the discrete space {0, 1} by some other space Z. DEFINITION 3.1 Let Z be a topological space with more than one point. A space X is Z- connected if and only if any continuous map from X to Z is constant. Note that in the above definition, Z is restricted to be a space with more than one point. Otherwise, the image of X is always constant and the definition makes no sense. As an immediate consequence, the following proposition can be proved. PROPOSITION 3.2 A Z-connected space is connected. Proof: Since Z has at least two points, there exists a continuous injection i such that i : {0, 1} Z Then for any continuous map f : X {0, 1}, io f is also a continuous function. Now X is Z-connected, by definition, io f is constant, thus f is constant. Therefore, X is connected. 9

15 This proposition ensures that the new Z-connectedness is stronger than original connectedness, and the definition of Z-connected spaces is suitable for the purpose of finding a stronger notion of connectedness. 3.2 EXAMPLES OF Z-CONNECTED SPACES To get a better understanding of Z-connected spaces, we need some examples. If a space X is Z-connected, the property of X depends greatly on the space Z. So we start our discussion with various space Z. There are two approaches: the connectedness of Z and the topological property of Z. First let's begin with the connectedness of Z. Roughly speaking, there are three possibilities: (1) Z is totally disconnected (2) Z is a collection of connected components, i.e. Z = {Z λ λ Λ } (3) Z is connected The following proposition shows that if Z is totally disconnected, the definition of Z- connectedness has the same meaning as the usual connectedness. PROPOSITION 3.3: If Z is totally disconnected, Z-connectedness is equivalent to connectedness. Proof: By proposition 2.2, if X is Z-connected, it is connected. Conversely, if X is connected and Z is totally disconnected, then for any continuous map f : X Z, f [X] is connected. However, the only connected subset of Z is onepoint space, so f is constant. Therefore, X is Z-connected. The next proposition shows that Z may be assumed to be connected. 10

16 PROPOSITION 3.4: Let Z be a space that is not totally disconnected and let Z = {Z λ λ Λ} be the collection of all connected components of Z. Then for any space X, X is Z- connected if and only if X is Z λ -connected for all λ Λ. Proof: Suppose X is Z-connected. By definition, any continuous map from X to Z is constant. For all λ Λ, the inclusion i λ : Z λ Z is an injective continuous map. For any continuous map f : X Z λ, iλo f : X Z is continuous and constant. Therefore f is constant, i.e. X is Z λ - connected. Conversely, suppose X is Z λ -connected for all λ Λ. Since Z is not totally disconnected, there exists at least a Z α having more than one point. X is Z α - connected, so by proposition 3.1, X is connected. For any continuous map g: X Z, g[x] is connected and belongs to some Z β. X is Z β -connected, so g is constant, X is Z-connected. Next, we shall see how the space X varies when different topologies are added to a twopoint set {0, 1}. Then there are only three types of topologies on Z, namely, indiscrete topology, order topology, and discrete topology. For simplicity, we write 2 i : the space {0, 1} with indiscrete topology, whose open sets are, {0, 1}; 2 o : the space {0, 1} with order topology, whose open sets are, {0}, {0, 1}; 2 d : the space {0, 1} with discrete topology, whose open sets are, {0}, {1}, {0, 1}; CLAIM: X is 2 i -connected, if and only if X is a one-point space. Proof: If X is a one-point space, for any continuous map f : X Z, f (X) is constant. Conversely, if X has more than one point, X = U V, where U and V nonempty and disjoint. Define f : X Z by ƒ [U] = 0, ƒ [V] = 1. This function is continuous but not constant. Thus X is not 2 i -connected. Therefore X is not Z connected except that X is one-point. CLAIM: X is 2 o -connected if and only if X is indiscrete. 11

17 Proof: If X is indiscrete, for any continuous map f from X to the space 2 o, {0} is open in the space 2 o, so f -1 (0) is open in X, thus f -1 (0) = X or. If f -1 (0) = X, ƒ(x) = 0; if f -1 (0) =, ƒ(x) = 1. In either case, ƒ is constant. Conversely, if X is not indiscrete, there exists a proper open set S of X. Define f : X {0, 1} by f [S] = 0, and f [X-S] = 1. Then f -1 (0) = S, f -1 ({0, 1}) = X, thus f is continuous but not constant, and hence not 2 o -connected. Therefore, X is 2 o -connected if and only if X is indiscrete. CLAIM: X is 2 d -connected if and only if X is connected. Proof: It is exactly the definition of a connected space. The following proposition is a summary of the above cases. PROPOSITION 3.5 Let Z be a two-point space. Then (I) X is 2 i -connected if and only if X is a one-point space. (II) X is 2 o -connected if and only if X is indiscrete. (III) X is 2 d -connected if and only if X is connected. This proposition can be generalized to finite case. For a finite space Z, it turns out that these are the only types of connectedness that arise. Recall separation axioms of a topological space, X is T 1 if for any distinct x, y X, there exists open neighborhoods U of x and V of y such that y is not in U and x is not in V. It is easy to see that X is T 1 if and only if every singleton set in X is closed. In the following studies, this property will be used very often. PROPOSITION 3.6 Let Z be a space that is not T 1. If X is Z-connected, then X is either indiscrete or is a one-point space. Proof: Let Z be a space that is not T 1, i.e. there exists a connected component Z λ that contains more than one point. Suppose X is neither one point nor indiscrete, then X has a proper open set. It is sufficient to prove that there exists a continuous map f : X 12

18 Z that is not constant. Define f : X Z by f [U] = x, f [X U] = y, where x and y are elements of Z λ. Note that the subspace {x, y} is either 2 i or 2 o but never 2 d, thus f is continuous and not constant. Therefore, X is either indiscrete or is a one-point space. This proposition tells us that whether Z is T 1 is crucial in generating nontrivial Z- connectedness. Note that a finite T1 space is discrete. The first nontrivial example of Z- connectedness can be constructed by taking Z to be the space Ζ of integers equipped with the complement finite topology. In fact, Ζ is the coarsest space which is infinite and T PROPERTIES OF Z-CONNECTEDNESS The concept of Z-connectedness is very similar to that of connectedness. Most of results for connected spaces still hold for Z-connected spaces, and they can be proved easily by considering continuous maps into the space Z. To make the Z-connectedness nontrivial, the space Z is assumed to be a T 1 space. For a connected space, a continuous image of a connected space is connected. And for a Z-connected space, this property still holds. PROPOSITION 3.7: A continuous image of Z-connected space is Z-connected. Proof: Let X be any Z-connected space. By definition, any continuous map from X to Z is constant. Let f : X Y be a continuous surjective map and g : Y Z be continuous. Then go f is continuous and constant, so g[y] is constant. Therefore, Y is Z-connected, i.e. the continuous image of X is Z-connected. If a collection of connected subspaces has a common point, then their union is connected. A similar statement can be proved for Z-connected spaces. 13

19 PROPOSITION 3.8: If { X α }is a collection of Z-connected subspaces of a space X such that α X α, then α X α is Z-connected. Proof: For any continuous map f: α X α Z, let map i α : X α α X α be the inclusion map and let f : α X α Z be any continuous map. Since X α is Z-connected, f X Æ o i α : Z is continuous and thus constant. And α X α, so there exist a p such that p α X α, i.e. p X α for all α. Then f Therefore, f is constant and α X α is Z-connected. o i α is constant and equals to f (p). If A is a connected subset of a space X, and A B A, then B is connected. In Z- connected spaces, this proposition can be written as follows. PROPOSITION 3.9: Let A and B be subsets in a space X such that A B A. If A is Z- connected, then B is Z-connected. Proof: Let f : B Z be any continuous map where A B A and let f A : A Z be the restriction of f. Since A is Z-connected, and f A is continuous, f A (A) = f (A) is constant. Z is a T 1 space, thus f (A) is closed. Note that B A = A B= B, therefore, B f( B) = f( A ) f( A) = f( A). Thus f (B) is constant and B is Z-connected. A Z-connected component containing p is defined as the largest Z-connected set in X which contains the point p. For such a Z-connected component, the following property can be proved. COROLLARY 3.10: Each Z-connected component of a space is closed. Proof: Let A be any Z-connected component, i.e. A is the largest Z-connected set in X containing p. By proposition 3.9, A is Z-connected and closed. Also, A is larger than A. Thus A = A, and a Z-connected component is closed. Just like the product of an arbitrary family of connected spaces is connected, the product of Z-connected spaces is Z-connected. 14

20 PROPOSITION 3.11: The topological product of an arbitrary family of Z-connected spaces is Z-connected. Proof: First, let's prove the product X Y of Z-connected spaces X and Y is Z- connected. Let (a, b) and (c, d) be any two points of X Y. Then X {} b and {} c Y are Z-connected and have the intersection (c, b). By proposition 3.8, X {} b {} c Y is Z-connected. Then the T-shaped space T = ( X {}) b ({ c} Y) is Z-connected. Y (a, b) (c, d) X Fig 2. A T-shaped space in XY plane. The space X Y is the union of all T-shaped spaces, and those spaces are Z- connected and have a common point ( ab, ). Thus X Y is Z-connected. Next, for any finite product of Z-connected spaces, X1 X2 L Xn is homeomorphic to ( X1 L X 1) X n n, and by induction, it is Z-connected. Finally, let's consider an arbitrary family {X α } of Z-connected spaces. Let X = Π X α and b be a given point in X. Define a subspace X( α1, L α n ) of X. It consists of all points ( x α ) such that xα = bα for α α, 1 L αn. X( α1, Lα n ) is homeomorphic with the finite product X X L X, and hence is Z-connected. Then we define a 1 2 n α α α subspace Y be the union of the above subspaces, i.e. Y = X( α1, L α n ), and Y is Z- connected since all of X( α1, L α n ) has a common point b. CLAIM: Y = X under the product topology. Proof of the claim: Let's take an arbitrary point ( x α ) of X and an arbitrary open neighborhood U = U α of ( x α ), and prove U intersects Y. 15

21 Each set U α is open in X α, and U α = X α, except for finite indices, say α = α, 1 L α. Construct a point ( y α ) of X by setting y α = x α for α = α, 1 L αn, and y α = b α for other values of α. Then ( y α ) is a point of Y since ( y α ) X ( α, 1 L αn n ). Also, it is a point of U, since y α = x α U α for all α = α 1, L α and y α = b α x α for all other values of α. Hence U intersects Y. This shows that Y Z-connected. n = X. As Y is Z-connected, X is A local concept of Z-connectedness is also possible. A space X is said to be locally Z- connected if it has a basis consisting of Z-connected open sets. A connected component of a locally connected space is open, and every open subspace of a locally connected space is locally connected. A Z-connected component has the similar property. PROPOSITION 3.12: Each Z-connected component of a locally Z-connected space is open. Proof: Suppose X is a locally Z-connected space with basis such that Bi, Bi is Z-connected. Let A be any Z-connected component of space X. Take any a A, then there exists some B, such that a B. By definition of Z-connected component, B A. Thus A is union of open sets and hence is open. PROPOSITION 3.13: Every open subspace of a locally Z-connected space is locally Z- connected. Proof: Suppose X is locally Z-connected and is basis of X such that every member of is Z-connected. Let A be an open subspace. Then ' = { B B A } is a basis of A consisting of Z-connected open sets. Therefore A is locally Z-connected. The locally Z-connectedness has another important property. That is, in a locally Z- connected space, the Z-connectedness equals connectedness. PROPOSITION 3.14: Suppose X is locally Z-connected. Then X is connected if and only if X is Z-connected. 16

22 Proof: If X is Z-connected, then X is connected by proposition 3.2. Conversely, suppose X is connected and locally Z-connected. Take a Z-connected component C of X, by 3.12, C is open in X, and by 3.10, C is open. As X is connected, C = X. Therefore, X is Z-connected. 17

23 CHAPTER STRONGLY CONNECTED SPACES CONCEPT OF STRONGLY CONNECTEDNESS In this chapter, the concept of strongly connected space will be introduced. Recall the second definition of connectedness, a space is connected if it has no separation. Now if the separation is extended from a pair of open sets U and V to countable sets, a similar but slightly stronger connectedness can be produced. But here we must use closed sets. The definition is given as follows. DEFINITION 4.1 A space X is strongly connected if and only if it is not a disjoint union of countably many but more than one closed set. Note the similarity between this definition and that of connectedness. If X is connected, and E 1 and E 2 are any two nonempty disjoint closed sets of X, then X E 1 E 2. If X is strongly connected, and E i are nonempty disjoint closed sets of X, then X E 1 E 2 E 3 Recall the other definition of connectedness, that is, a space X is connected if and only if any continuous map from it to the discrete space is constant. It is natural to seek a similar definition for strongly connected spaces. 18

24 As mentioned last chapter, the first nontrivial example of Z-connected space is constructed by taking Z to be the space of integers equipped with the complement finite topology, denoted by Ζ. It can be proved that a strongly connected space defined above is equivalent to the Ζ-connectedness. This fact is presented in the following definition. DEFINITION 4.2 A space is strongly connected if and only if it is Ζ-connected. Proof: Suppose X is strongly connected. Let f : X Ζ be any continuous map. CLAIM: A continuous image of a strongly connected space is strongly connected. Proof of the claim: Using the above assumption, and further assume f is surjective. Suppose f(x) is not strongly connected, by definition it is a disjoint union of countably many but more than one closed sets. Since f is continuous, and the inverse image of closed sets are still closed, X is also a disjoint union of closed sets. Therefore, f(x) is strongly connected. By claim, f(x) is strongly connected. The only strongly connected subset of Ζ are the one-point spaces. Hence f is constant, i.e. X is Ζ-connected. Conversely, suppose X is a disjoint union of countably many but more than one closed sets, X = E i. Then define f : X Ζ by taking f(x) = i whenever x E i. This f is continuous and not constant. So X is not Ζ-connected. Therefore, X is strongly connected if and only if it is Ζ-connected. The similarity between this definition and that of connectedness is obvious. For any space X and any continuous map f, f : X 2 d is constant X is connected; f : X Ζ is constant X is strongly connected. 4.2 USUAL PROPERTIES FOR STRONGLY CONNECTED SPACES From the second definition of strongly connectedness, a strongly connected space is nothing but a special case of Z-connectedness. Thus, all the properties proved for Z- 19

25 connected spaces in the second chapter definitely apply to strongly connected spaces. At the same time, it has many other interesting properties. The properties, as a Z-connected space owns, can be summarized in the following proposition, and the proofs are omitted, since they are exactly the same as the last chapter. PROPOSITION 4.3 Let X be a strongly connected space, then the following is true. (1) Any continuous image of X is strongly connected. (2) The union of any family of strongly connected sets with a common point is strongly connected. (3) Let A and B be subsets in a space X such that A B A, where A is the closure of A. If A is strongly connected, then B is strongly connected. (4) For each point p in a space X, the strongly connected component C(p) of X is closed. (5) The topological product of an arbitrary family of strongly connected spaces is strongly connected. (6) Every open subspace of a locally strongly connected space is locally strongly connected. The component of a locally strongly connected space is open. 4.3 STRONGLY CONNECTEDNESS IN COMPACT SPACES Strongly connectedness is a stronger notion of connectedness. In another word, given a connected space, we can make it strongly connected by adding some conditions. But what conditions should be added is the difficulty. Our starting point is a connected spaces, thus a continuum may be useful. The concept of a continuum is defined on a connected set. DEFINITION 4.4 A compact connected set is called a continuum. 20

26 Let int(a) represents the interior of A, Cl(A) be closure of A, and bd(a) be the boundary of A. Then a continuum has the following property. PROPOSITION 4.5 If A is any continuum in a Hausdorff space and B is any open set such that A B A (X B), then every component of A Cl(B) intersects bd(b). Proof: The proof can be found in [2], page Using this result, we can prove that a Hausdorff continuum is strongly connected. Or said differently, PROPOSITION 4.6 Let X be a compact Hausdorff space. Then X is connected if and only if X is strongly connected. Proof: If X is strongly connected, then X is connected. Conversely, if X is a compact Hausdorff connected space, and if it is not strongly connected, then by definition, X is a union of a countably many but more than one disjoint closed sets. X = K i, where K i are closed disjoint sets. Since a compact Hausdorff space is normal, X is a normal space. We can find an open set G 1 such that K2 G 1, and Cl(G1) K 1 =. Let X 1 be a component of Cl(G 1 ) which intersects K 2. Then X 1 is compact and connected. Now X1 bd(g 1 ), i.e. X1 contains a point p bd(g 1 ) such that p G 1 and p K 1. Hence X1 intersects some K i for i > 2. Let Kn2 Kn2 be the first K i for i >2 which intersects X1, and let G 2 be an open set satisfying G 2, and Cl(G 2 ) K 2 =. Then let X 2 be a component of X 1 Cl(G 2 ) which contains a point of K n2. Again we have X 2 bd(g2), and X 2 contains some point p bd(g 2 ) such that p G 2, p K 1 K 2. Hence X 2 intersects some K i for i > n 2, and X 2 K i = for i < n 2. Let Kn3 be the first K i for i > n 2, which intersects X 2, then by methods similar to the above we can find a compact connected X 3 such that X 3 X 2 X 1, and X 3 intersects some K i with i > n 3 but X 3 K i = for i < n 3. 21

27 In this manner, we obtain a sequence of subcontinua of X: X 1 X 2 X 3, such that for each j, X j K i = for i < n j and n j as j. we know that X i. Also, ( X i ) K j = for all j, so that ( X i ) ( K j )= or ( X i ) X =. But ( X i ) X, which contradicts the fact that X i is nonempty. Therefore, if X is a compact Hausdorff space, it is connected if and only if it is strongly connected. We know that connectedness and local connectedness are similar in some manner, thus it is natural to think of the above proposition in a local way. Indeed, a local version is also true. PROPOSITION 4.7 Let X be a locally compact Hausdorff space. Suppose X is locally connected. Then X is locally strongly connected. Proof: Let O be an open neighborhood of a point x X. Then there exists a compact neighborhood V of x lying inside O. Let C be a connected component of V containing x. Since V is a neighborhood of x and X is locally connected, C is a neighborhood of x. Since C is closed in V and V is compact, C is compact. C is a compact connected neighborhood of x lying inside O. By proposition 4.6, C is strongly connected. There is one way to combine proposition 4.6 and 4.7, and it is stated in the next proposition. PROPOSITION 4.8 Let X be a locally compact Hausdorff space. Suppose X is locally connected and connected. Then X is strongly connected. Proof: This follows from proposition 3.10 and proposition 4.7. COROLLARY 4.9 A strongly connected T 1 space having more than one point is uncountable. Proof: A one-point set in a T 1 space is closed, thus by the definition of strongly connected space, a T 1 space cannot have countably many but more than one point. 22

28 COROLLARY 4.10 The space n is uncountable. Proof: is strongly connected, and by proposition 4.3.5, corollary 4.9, it is uncountable. n is strongly connected. By Note that a continuum is different from a connected space essentially because it is compact. A similar topological property to compactness is complete metric, thus we continue our discussion with complete metric spaces. 4.4 STRONGLY CONNECTEDNESS IN COMPLETE METRIC SPACES In a metric space, a Cauchy sequence is a sequence (x n ) of points in X such that for every positive real number ε, there exists a positive integer n satisfying d [x i, x j ] < ε for all integers i > n and j > n. A metric space X is said to be complete if and only if every Cauchy sequence in X converges to some point in X. In a complete metric space, the following property can be shown. PROPOSITION 4.11 A connected locally connected complete metric space is strongly connected. Proof: Let X be a connected locally connected complete metric space. Suppose that X is a countable disjoint union E i, where E i is closed. Since X is connected, we may assume that each E i is nonempty. Let k be a positive integer. Since X is a locally connected metric space, the collection Χ k of all connected open sets of diameter less than 1 k is a basis for the topology of X. Let ϑ 1 = {O Χ 1 O X \ E 1 }. Then X \ E 1 = i= 2 E i = o I O. 2 CLAIM: One of the member in ϑ 1 intersects infinitely many E i s for i > 1. 23

29 Proof of the claim: Suppose it is not true, then each of these connected open sets in ϑ 1 lie in some E i for some i >1. Thus each E i is a union of some of these open sets and hence is open. This contradicts the fact that X is connected. Let O 1 be a member in ϑ 1 which intersects infinitely many E i s for i >1. In particular, this implies that O 1 \ E 2 is nonempty. Next let ϑ 2 = {O Χ 2 O O 1 \ E 2 }. Then O 1 \ E 2 = O. The connectedness of 2 O 1 implies that there exists a member O 2 in ϑ 2 which intersects infinitely many E i s for i > 2. Therefore O 2 \ E 3 is nonempty. Continuing in this way produces a decreasing sequence O 1 O 2 O 3 of nonempty connected open subsets of X such that for each positive integer k, O k Χ k, O k O k-1 \ E k. Here O 0 is taken to be X. Now for each positive integer k, pick an element x k O k. Since the diameter of O k is less than 1 k, (x k) is a Cauchy sequence. By the completeness of X, (x k ) converges to an element x X. Hence X lies in some E n. But x also lies in O n, this contradicts the fact that O n is disjoint from E n. Consequently, x cannot be expressed as a disjoint union of countably many but more than one nonempty closed subsets. This shows that X is strongly connected. o I 4.5 EXAMPLES OF STRONGLY CONNECTED SPACES In this section, we emphasis on those spaces that are connected but not strongly connected, since these spaces help us to differentiate connectedness and strongly connectedness. The first example is the space Ζ. It is connected since it has no separation, and it is not strongly connected since the identity map from Ζ to itself is not constant. The second example is given in [1], and it is all the rational points in the plane on or above the x-axis with following topology. If (a, b) X, then the neighborhood of (a, b) is 24

30 b b ε < + < 3 3 { ( ) } ( ) ε ( ) 2 2 ab, r,0 : r a r,0 : r a And its graph representation is y (a, b) a+ b 3 ( ) -ε +ε ( ) -ε +ε x Fig 3. graph representation for example (2). This example is not strongly connected, since the rational points are countable, and as a subset of 2, the space is also countable. It is T 1 since it is Hausdorff. Thus by corollary 4.9, it is not strongly connected. Its connectedness follows from the next claim. CLAIM: If any pair of neighborhood of a space has a common limit point, the space is connected. Proof of the claim: Suppose the space X is not connected, then it has a separation U V. U and V are both open and closed, thus they do not have a common limit point. Therefore, X must be connected. For this example, the closure of any neighborhood is y 25

31 (a, b) a+ b 3 ( ) -ε +ε ( ) -ε +ε x Fig 4. graph for the closure of an open set Thus any two neighborhood has at least one point in common, i.e. in the following graph, A 1 meets A 2 at some point. y A 1 A 2 (a, b) ( ) -ε +ε ( ) -ε +ε x Fig 5. two neighborhood always meet The last example can be found in [5], and it is interesting because it is not easy to find a connected subspace in 2 that is not strongly connected. It is the union of n+ 1 2 (i) xn { (2, y) 0 y 1} =, n = 0, 1, 2 26

32 (ii) (ιιι) yn = (0, y) < y< 2 n+ 1 2 n z n The graph for it is n+ 1 n+ 1 2 π = (2 cos θ,2 sin θ) θ 2π 2 Fig 6. graph representation for example (3). Note that cn = xn yn znis connected, and X n =U c. n n= 0 It is clearly a disjoint union of closed set, and hence not strongly connected. To prove its connectedness, we take an arbitrary nonempty subset U that is both open and closed in X, and show this U = X. First of all, U is nonempty, so there is at least one point p U. Then p c n for some n. U is an open neighborhood of p, and c n is connected, thus c n lies completely in U. 27

33 Next, consider the point (0, 2 -n+1 ), it is a point of c n and thus lies in U. Take an open disk around the point, then it intersects all of x n for n greater than some N. Therefore, c n U for all n > N for some number N. Since the whole set y n are the limit points of x n for n greater than N, y n U for all n. Thus c n U for all n. Therefore, U = c n = X, and X is connected. 28

34 CHAPTER CONCLUSION So far, we have discussed three kind of connectedness, the usual connectedness, the Z- connectedness and the strongly connectedness, and the strongly connectedness finally meets our purpose of finding a stronger notion of connectedness. 5.1 SUMMARY OF THE WORK In this report, we first review the concept of connectedness. It is a well-developed concept, and its definition, properties and examples are all ready to use. Thus, a quick revision enables us doing further study. By slightly modifying its definition, we gain a Z-connected space, which has almost the same features as a connect space. The Z-connectedness may vary greatly, from the usual connectedness to one-point connectedness, therefore, being a general Z-connected space, it does not have anything very interesting. However, a specific Z-connected space -- the strongly connected space is an ideal notion of stronger connectedness. Since the space Z is carefully chosen, the strongly connectedness is only a little stronger than the usual connectedness. We have shown that the strongly connectedness not only has the same property as the Z-connected space, but also has many new properties. Moreover, it is also mentioned in the report that how to make a connected space strongly connected. 29

35 5.2 FURTHER SUGGESTIONS The Z-connected space is an intermediate product of the project, but it has many interesting properties. It might be worthwhile to continue the research, for example, we can choose another space Z other than the one used in strongly connected spaces. The other point to note is that the conditions for a connected space to be strongly connected that presented in this report is still too strict, thus finding more suitable conditions will be a challenge. 30

36 REFERENCE: [1] Bing, R. H. (1953), A connected countable Hausdorff space, Proc, A.M.S. [2] Duda, E. & Whyburn, G. (1978), Dynamic Topology, Undergraduate texts in Mathematics, Springer_Berlag. [3] Hocking, J. G. & Young, G. S. (1988), Topology, Addison-Wesley Publishing Company, Inc. [4] Hu, S.T. (1966), Introduction to general topology, Holden-Day inc. [5] Kuratowski, K. (1948), Topologie, Vol 1, Warsaw. [6] Munkers, J. R. (1975), Topology, a first course, Rrentice-Hall inc. New Jersey. [7] Robert B, Ash. (1993), Real variables with basic metric space topology, New York. [8] Smullyan, R. M. (1969), The continuum hypothesis, the mathematical Sciences, A Collection of Essays, The M.I.T. Press, Cambridge. [9] Steen, L. A. (1970), Counterexamples in Topology, Holt, Rinehart & Winston Inc, New York. [10] Willard, S. (1970), General Topology, Addison-Wesley Publishing Company, Inc. 31

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

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

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

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

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

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

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

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

MA651 Topology. Lecture 9. Compactness 2.

MA651 Topology. Lecture 9. Compactness 2. MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology

More information

MAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.

MAS3706 Topology. Revision Lectures, May I do not answer  enquiries as to what material will be in the exam. MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar

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

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

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

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

Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together

More information

Problem Set 2: Solutions Math 201A: Fall 2016

Problem Set 2: Solutions Math 201A: Fall 2016 Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that

More 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

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

Solve EACH of the exercises 1-3

Solve EACH of the exercises 1-3 Topology Ph.D. Entrance Exam, August 2011 Write a solution of each exercise on a separate page. Solve EACH of the exercises 1-3 Ex. 1. Let X and Y be Hausdorff topological spaces and let f: X Y be continuous.

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

Some Background Material

Some Background Material Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

More information

Math 201 Topology I. Lecture notes of Prof. Hicham Gebran

Math 201 Topology I. Lecture notes of Prof. Hicham Gebran Math 201 Topology I Lecture notes of Prof. Hicham Gebran hicham.gebran@yahoo.com Lebanese University, Fanar, Fall 2015-2016 http://fs2.ul.edu.lb/math http://hichamgebran.wordpress.com 2 Introduction and

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

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

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

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

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

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

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered

More information

Chapter 2 Metric Spaces

Chapter 2 Metric Spaces Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics

More information

2. The Concept of Convergence: Ultrafilters and Nets

2. The Concept of Convergence: Ultrafilters and Nets 2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two

More information

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X. Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

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

CARDINALITY OF THE SET OF REAL FUNCTIONS WITH A GIVEN CONTINUITY SET

CARDINALITY OF THE SET OF REAL FUNCTIONS WITH A GIVEN CONTINUITY SET CARDINALITY OF THE SET OF REAL FUNCTIONS WITH A GIVEN CONTINUITY SET JIAMING CHEN AND SAM SMITH Abstract. Expanding on an old result of W. H. Young, we determine the cardinality of the set of functions

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

Axioms of separation

Axioms of separation Axioms of separation These notes discuss the same topic as Sections 31, 32, 33, 34, 35, and also 7, 10 of Munkres book. Some notions (hereditarily normal, perfectly normal, collectionwise normal, monotonically

More 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

s P = f(ξ n )(x i x i 1 ). i=1

s P = f(ξ n )(x i x i 1 ). i=1 Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological

More 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

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

2 Sequences, Continuity, and Limits

2 Sequences, Continuity, and Limits 2 Sequences, Continuity, and Limits In this chapter, we introduce the fundamental notions of continuity and limit of a real-valued function of two variables. As in ACICARA, the definitions as well as proofs

More information

Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

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

ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET

ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET Novi Sad J. Math. Vol. 40, No. 2, 2010, 7-16 ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET M.K. Bose 1, Ajoy Mukharjee 2 Abstract Considering a countable number of topologies on a set X, we introduce the

More information

Sanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle

Sanjay Mishra. Topology. Dr. Sanjay Mishra. A Profound Subtitle Topology A Profound Subtitle Dr. Copyright c 2017 Contents I General Topology 1 Compactness of Topological Space............................ 7 1.1 Introduction 7 1.2 Compact Space 7 1.2.1 Compact Space.................................................

More 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

A LITTLE REAL ANALYSIS AND TOPOLOGY

A LITTLE REAL ANALYSIS AND TOPOLOGY A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set

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

5 Set Operations, Functions, and Counting

5 Set Operations, Functions, and Counting 5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,

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

3 Hausdorff and Connected Spaces

3 Hausdorff and Connected Spaces 3 Hausdorff and Connected Spaces In this chapter we address the question of when two spaces are homeomorphic. This is done by examining two properties that are shared by any pair of homeomorphic spaces.

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

2. Metric Spaces. 2.1 Definitions etc.

2. Metric Spaces. 2.1 Definitions etc. 2. Metric Spaces 2.1 Definitions etc. The procedure in Section for regarding R as a topological space may be generalized to many other sets in which there is some kind of distance (formally, sets with

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

CHODOUNSKY, DAVID, M.A. Relative Topological Properties. (2006) Directed by Dr. Jerry Vaughan. 48pp.

CHODOUNSKY, DAVID, M.A. Relative Topological Properties. (2006) Directed by Dr. Jerry Vaughan. 48pp. CHODOUNSKY, DAVID, M.A. Relative Topological Properties. (2006) Directed by Dr. Jerry Vaughan. 48pp. In this thesis we study the concepts of relative topological properties and give some basic facts and

More information

F 1 =. Setting F 1 = F i0 we have that. j=1 F i j

F 1 =. Setting F 1 = F i0 we have that. j=1 F i j Topology Exercise Sheet 5 Prof. Dr. Alessandro Sisto Due to 28 March Question 1: Let T be the following topology on the real line R: T ; for each finite set F R, we declare R F T. (a) Check that T is a

More information

Solutions to Tutorial 8 (Week 9)

Solutions to Tutorial 8 (Week 9) The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/

More information

CHAPTER 5. The Topology of R. 1. Open and Closed Sets

CHAPTER 5. The Topology of R. 1. Open and Closed Sets CHAPTER 5 The Topology of R 1. Open and Closed Sets DEFINITION 5.1. A set G Ω R is open if for every x 2 G there is an " > 0 such that (x ", x + ") Ω G. A set F Ω R is closed if F c is open. The idea is

More information

VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES Bull. Austral. Math. Soc. 78 (2008), 487 495 doi:10.1017/s0004972708000877 VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES CAROLYN E. MCPHAIL and SIDNEY A. MORRIS (Received 3 March 2008) Abstract

More information

Introduction to Dynamical Systems

Introduction to Dynamical Systems Introduction to Dynamical Systems France-Kosovo Undergraduate Research School of Mathematics March 2017 This introduction to dynamical systems was a course given at the march 2017 edition of the France

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

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

Theorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers

Theorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower

More information

Exam 2 extra practice problems

Exam 2 extra practice problems Exam 2 extra practice problems (1) If (X, d) is connected and f : X R is a continuous function such that f(x) = 1 for all x X, show that f must be constant. Solution: Since f(x) = 1 for every x X, either

More information

TOPOLOGY TAKE-HOME CLAY SHONKWILER

TOPOLOGY TAKE-HOME CLAY SHONKWILER TOPOLOGY TAKE-HOME CLAY SHONKWILER 1. The Discrete Topology Let Y = {0, 1} have the discrete topology. Show that for any topological space X the following are equivalent. (a) X has the discrete topology.

More information

MAS331: Metric Spaces Problems on Chapter 1

MAS331: Metric Spaces Problems on Chapter 1 MAS331: Metric Spaces Problems on Chapter 1 1. In R 3, find d 1 ((3, 1, 4), (2, 7, 1)), d 2 ((3, 1, 4), (2, 7, 1)) and d ((3, 1, 4), (2, 7, 1)). 2. In R 4, show that d 1 ((4, 4, 4, 6), (0, 0, 0, 0)) =

More information

(c) For each α R \ {0}, the mapping x αx is a homeomorphism of X.

(c) For each α R \ {0}, the mapping x αx is a homeomorphism of X. A short account of topological vector spaces Normed spaces, and especially Banach spaces, are basic ambient spaces in Infinite- Dimensional Analysis. However, there are situations in which it is necessary

More information

POL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005

POL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005 POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1

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

Chapter 1. Sets and Mappings

Chapter 1. Sets and Mappings Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write

More 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

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

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

Austin Mohr Math 730 Homework. f(x) = y for some x λ Λ

Austin Mohr Math 730 Homework. f(x) = y for some x λ Λ Austin Mohr Math 730 Homework In the following problems, let Λ be an indexing set and let A and B λ for λ Λ be arbitrary sets. Problem 1B1 ( ) Show A B λ = (A B λ ). λ Λ λ Λ Proof. ( ) x A B λ λ Λ x A

More information

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous: MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is

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

2 Topology of a Metric Space

2 Topology of a Metric Space 2 Topology of a Metric Space The real number system has two types of properties. The first type are algebraic properties, dealing with addition, multiplication and so on. The other type, called topological

More information

2.23 Theorem. Let A and B be sets in a metric space. If A B, then L(A) L(B).

2.23 Theorem. Let A and B be sets in a metric space. If A B, then L(A) L(B). 2.23 Theorem. Let A and B be sets in a metric space. If A B, then L(A) L(B). 2.24 Theorem. Let A and B be sets in a metric space. Then L(A B) = L(A) L(B). It is worth noting that you can t replace union

More information

Sets, Functions and Metric Spaces

Sets, Functions and Metric Spaces Chapter 14 Sets, Functions and Metric Spaces 14.1 Functions and sets 14.1.1 The function concept Definition 14.1 Let us consider two sets A and B whose elements may be any objects whatsoever. Suppose that

More 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

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

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

Principles of Real Analysis I Fall I. The Real Number System

Principles of Real Analysis I Fall I. The Real Number System 21-355 Principles of Real Analysis I Fall 2004 I. The Real Number System The main goal of this course is to develop the theory of real-valued functions of one real variable in a systematic and rigorous

More information

Sets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions

Sets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions Sets and Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Notation x A means that element x is a member of set A. x / A means that x is not a member of 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

MAT327 Big List. Ivan Khatchatourian Department of Mathematics University of Toronto. August 20, 2018

MAT327 Big List. Ivan Khatchatourian Department of Mathematics University of Toronto. August 20, 2018 MAT327 Big List Ivan Khatchatourian Department of Mathematics University of Toronto August 20, 2018 This is a large, constantly growing list of problems in basic point set topology. This list will include

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

Topological properties of Z p and Q p and Euclidean models

Topological properties of Z p and Q p and Euclidean models Topological properties of Z p and Q p and Euclidean models Samuel Trautwein, Esther Röder, Giorgio Barozzi November 3, 20 Topology of Q p vs Topology of R Both R and Q p are normed fields and complete

More information

*Room 3.13 in Herschel Building

*Room 3.13 in Herschel Building MAS3706: Topology Dr. Zinaida Lykova School of Mathematics, Statistics and Physics Newcastle University *Room 3.13 in Herschel Building These lectures concern metric and topological spaces and continuous

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 1. I. Foundational material

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 1. I. Foundational material SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 1 Fall 2014 I. Foundational material I.1 : Basic set theory Problems from Munkres, 9, p. 64 2. (a (c For each of the first three parts, choose a 1 1 correspondence

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

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

More information

CW-complexes. Stephen A. Mitchell. November 1997

CW-complexes. Stephen A. Mitchell. November 1997 CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,

More information

Maximilian GANSTER. appeared in: Soochow J. Math. 15 (1) (1989),

Maximilian GANSTER. appeared in: Soochow J. Math. 15 (1) (1989), A NOTE ON STRONGLY LINDELÖF SPACES Maximilian GANSTER appeared in: Soochow J. Math. 15 (1) (1989), 99 104. Abstract Recently a new class of topological spaces, called strongly Lindelöf spaces, has been

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

THE ANTISYMMETRY BETWEENNESS AXIOM AND HAUSDORFF CONTINUA

THE ANTISYMMETRY BETWEENNESS AXIOM AND HAUSDORFF CONTINUA http://topology.auburn.edu/tp/ http://topology.nipissingu.ca/tp/ TOPOLOGY PROCEEDINGS Volume 45 (2015) Pages 1-27 E-Published on October xx, 2014 THE ANTISYMMETRY BETWEENNESS AXIOM AND HAUSDORFF CONTINUA

More information

Math 4603: Advanced Calculus I, Summer 2016 University of Minnesota Notes on Cardinality of Sets

Math 4603: Advanced Calculus I, Summer 2016 University of Minnesota Notes on Cardinality of Sets Math 4603: Advanced Calculus I, Summer 2016 University of Minnesota Notes on Cardinality of Sets Introduction In this short article, we will describe some basic notions on cardinality of sets. Given two

More information

COUNTABLY S-CLOSED SPACES

COUNTABLY S-CLOSED SPACES COUNTABLY S-CLOSED SPACES Karin DLASKA, Nurettin ERGUN and Maximilian GANSTER Abstract In this paper we introduce the class of countably S-closed spaces which lies between the familiar classes of S-closed

More information

1 Directional Derivatives and Differentiability

1 Directional Derivatives and Differentiability Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=

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