Introduction to Topology Past Paper PDF (Renzo's Math 490, Winter 2007)

Renzo’s Math 490 Introduction to Topology Tom Babinec Chris Best Michael Bliss Nikolai Brendler Eric Fu Adriane Fung Tyler Klein Alex Larson Topcue Lee John Madonna Joel Mousseau Nick Posavetz Matt Rosenberg Danielle Rogers Andrew Sardon...

Renzo’s Math 490 Introduction to Topology Tom Babinec Chris Best Michael Bliss Nikolai Brendler Eric Fu Adriane Fung Tyler Klein Alex Larson Topcue Lee John Madonna Joel Mousseau Nick Posavetz Matt Rosenberg Danielle Rogers Andrew Sardone Justin Shaler Smrithi Srinivasan Pete Troyan Jackson Yim Elizabeth Uible Derek Van Farowe Paige Warmker Zheng Wu Nina Zhang Winter 2007 Mathematics 490 – Introduction to Topology Winter 2007 2 Contents 1 Topology 9 1.1 Metric Spaces...................................... 9 1.2 Open Sets (in a metric space)............................. 10 1.3 Closed Sets (in a metric space)............................ 11 1.4 Topological Spaces................................... 11 1.5 Closed Sets (Revisited)................................. 12 1.6 Continuity........................................ 13 1.7 Homeomorphisms.................................... 14 1.8 Homeomorphism Examples.............................. 16 1.9 Theorems On Homeomorphism............................ 18 1.10 Homeomorphisms Between Letters of Alphabet................... 19 1.10.1 Topological Invariants............................. 19 1.10.2 Vertices..................................... 19 1.10.3 Holes...................................... 20 1.11 Classification of Letters................................ 21 1.11.1 The curious case of the “Q”.......................... 22 1.12 Topological Invariants................................. 23 1.12.1 Hausdorff Property............................... 23 1.12.2 Compactness Property............................. 24 1.12.3 Connectedness and Path Connectedness Properties............. 25 2 Making New Spaces From Old 27 2.1 Cartesian Products of Spaces............................. 27 2.2 The Product Topology................................. 28 2.3 Properties of Product Spaces............................. 29 3 Mathematics 490 – Introduction to Topology Winter 2007 2.4 Identification Spaces.................................. 30 2.5 Group Actions and Quotient Spaces......................... 34 3 First Topological Invariants 37 3.1 Introduction....................................... 37 3.2 Compactness...................................... 37 3.2.1 Preliminary Ideas................................ 37 3.2.2 The Notion of Compactness.......................... 40 3.3 Some Theorems on Compactness........................... 43 3.4 Hausdorff Spaces.................................... 47 3.5 T1 Spaces........................................ 49 3.6 Compactification.................................... 50 3.6.1 Motivation................................... 50 3.6.2 One-Point Compactification.......................... 50 3.6.3 Theorems.................................... 51 3.6.4 Examples.................................... 55 3.7 Connectedness..................................... 57 3.7.1 Introduction.................................. 57 3.7.2 Connectedness................................. 58 3.7.3 Path-Connectedness.............................. 61 4 Surfaces 63 4.1 Surfaces......................................... 63 4.2 The Projective Plane.................................. 63 4.2.1 RP2 as lines in R3 or a sphere with antipodal points identified....... 63 4.2.2 The Projective Plane as a Quotient Space of the Sphere.......... 65 4.2.3 The Projective Plane as an identification space of a disc.......... 66 4.2.4 Non-Orientability of the Projective Plane.................. 69 4.3 Polygons......................................... 69 4.3.1 Bigons...................................... 71 4.3.2 Rectangles................................... 72 4.3.3 Working with and simplifying polygons................... 74 4.4 Orientability...................................... 76 4.4.1 Definition.................................... 76 4 Mathematics 490 – Introduction to Topology Winter 2007 4.4.2 Applications To Common Surfaces...................... 77 4.4.3 Conclusion................................... 80 4.5 Euler Characteristic.................................. 80 4.5.1 Requirements.................................. 80 4.5.2 Computation.................................. 81 4.5.3 Usefulness.................................... 83 4.5.4 Use in identification polygons......................... 83 4.6 Connected Sums.................................... 85 4.6.1 Definition.................................... 85 4.6.2 Well-definedness................................ 85 4.6.3 Examples.................................... 87 4.6.4 RP2 #T= RP2 #RP2 #RP2........................... 88 4.6.5 Associativity.................................. 90 4.6.6 Effect on Euler Characteristic......................... 90 4.7 Classification Theorem................................. 92 4.7.1 Equivalent definitions............................. 92 4.7.2 Proof...................................... 93 5 Homotopy and the Fundamental Group 97 5.1 Homotopy of functions................................. 97 5.2 The Fundamental Group................................ 100 5.2.1 Free Groups.................................. 100 5.2.2 Graphic Representation of Free Group.................... 101 5.2.3 Presentation Of A Group........................... 103 5.2.4 The Fundamental Group............................ 103 5.3 Homotopy Equivalence between Spaces........................ 105 5.3.1 Homeomorphism vs. Homotopy Equivalence................. 105 5.3.2 Equivalence Relation.............................. 106 5.3.3 On the usefulness of Homotopy Equivalence................. 106 5.3.4 Simple-Connectedness and Contractible spaces............... 107 5.4 Retractions....................................... 108 5.4.1 Examples of Retractions............................ 108 5 Mathematics 490 – Introduction to Topology Winter 2007 5.5 Computing the Fundamental Groups of Surfaces: The Seifert-Van Kampen The- orem........................................... 110 5.5.1 Examples:.................................... 112 5.6 Covering Spaces.................................... 113 5.6.1 Lifting...................................... 117 6 Mathematics 490 – Introduction to Topology Winter 2007 What is this? This is a collection of topology notes compiled by Math 490 topology students at the University of Michigan in the Winter 2007 semester. Introductory topics of point-set and algebraic topology are covered in a series of five chapters. Foreword (for the random person stumbling upon this document) What you are looking at, my random reader, is not a topology textbook. It is not the lecture notes of my topology class either, but rather my student’s free interpretation of it. Well, I should use the word free with a little bit of caution, since they *had to* do this as their final project. These notes are organized and reflect tastes and choices of my students. I have done only a minimal amount of editing, to give a certain unity to the manuscript and to scrap out some mistakes - and so I don’t claim merits for this work but to have lead my already great students through this semester long adventure discovering a little bit of topology. Foreword (for my students) Well, guys, here it is! You’ve done it all, and here is a semester worth of labor, studying, but hopefully fun as well. I hope every once in a while you might enjoy flipping through the pages of this book and reminiscing topology...and that in half a century or so you might be telling exaggerated stories to your grandchildren about this class. A great thank to you all for a very good semester! 7 Mathematics 490 – Introduction to Topology Winter 2007 8 Chapter 1 Topology To understand what a topological space is, there are a number of definitions and issues that we need to address first. Namely, we will discuss metric spaces, open sets, and closed sets. Once we have an idea of these terms, we will have the vocabulary to define a topology. The definition of topology will also give us a more generalized notion of the meaning of open and closed sets. 1.1 Metric Spaces Definition 1.1.1. A metric space is a set X where we have a notion of distance. That is, if x, y ∈ X, then d(x, y) is the “distance” between x and y. The particular distance function must satisfy the following conditions: 1. d(x, y) ≥ 0 for all x, y ∈ X 2. d(x, y) = 0 if and only if x = y 3. d(x, y)=d(y, x) 4. d(x, z) ≤ d(x, y) + d(y, z) To understand this concept, it is helpful to consider a few examples of what does and does not constitute a distance function for a metric space. Example 1.1.2. For any space X, let d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. This metric, called the discrete metric, satisfies the conditions one through four. Example 1.1.3. The Pythagorean Theorem gives the most familiar notion of distance for points in Rn. In particular, when given x = (x1 , x2 ,..., xn ) and y = (y1 , y2 ,..., yn ), the distance f as v u n uX d(x, y) = t (xi − yi )2 i=1 9 Mathematics 490 – Introduction to Topology Winter 2007 Example 1.1.4. Suppose f and g are functions in a space X = {f : [0, 1] → R}. Does d(f, g) =max|f − g| define a metric? Again, in order to check that d(f, g) is a metric, we must check that this function satisfies the above criteria. But in this case property number 2 does not hold, as can be shown by considering two arbitrary functions at any point within the interval [0, 1]. If |f (x) − g(x)| = 0, this does not imply that f = g because f and g could intersect at one, and only one, point. Therefore, d(f, g) is not a metric in the given space. 1.2 Open Sets (in a metric space) Now that we have a notion of distance, we can define what it means to be an open set in a metric space. Definition 1.2.1. Let X be a metric space. A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x, y) < r}. We recognize that this ball encompasses all points whose distance is less than r from x. Definition 1.2.2. A subset O ⊆ X is open if for every point x ∈ O, there is a ball around x entirely contained in O. Example 1.2.3. Let X = [0, 1]. The interval (0, 1/2) is open in X. Example 1.2.4. Let X = R. The interval [0, 1/2) is not open in X. With an open set, we should be able to pick any point within the set, take an infinitesimal step in any direction within our given space, and find another point within the open set. In the first example, we can take any point 0 < x < 1/2 and find a point to the left or right of it, within the space [0, 1], that also is in the open set [0, 1). However, this cannot be done with the second example. For instance, if we take the point within the set [0, 1), say 0, and take an infinitesimal step to the left while staying within our given space X, we are no longer within the set [0, 1). Therefore, this would not be an open set within R. If a set is not open, this does not imply that it is closed. Furthermore, there exists sets that are neither open, nor closed, and sets that are open and closed. Lastly, open sets in spaces X have the following properties: 1. The empty set is open 2. The whole space X is open 3. The union of any collection of open sets is open 4. The intersection of any finite number of open sets is open. 10 Mathematics 490 – Introduction to Topology Winter 2007 1.3 Closed Sets (in a metric space) While we can and will define a closed sets by using the definition of open sets, we first define it using the notion of a limit point. Definition 1.3.1. A point z is a limit point for a set A if every open set U containing z intersects A in a point other than z. Notice, the point z could be in A or it might not be in A. The following example will help make this clear. Example 1.3.2. Consider the open unit disk D = {(x, y) : x2 + y 2 < 1}. Any point in D is a limit point of D. Take (0, 0) in D. Any open set U about this point will contain other points in D. Now consider (1, 0), which is not in D. This is still a limit point because any open set about (1, 0) will intersect the disk D. The following theorem and examples will give us a useful way to define closed sets, and will also prove to be very helpful when proving that sets are open as well. Definition 1.3.3. A set C is a closed set if and only if it contains all of its limit points. Example 1.3.4. The unit disk in the previous example is not closed because it does not contain all of its limit points; namely, (1, 0). Example 1.3.5. Let A = Z, a subset of R. This is a closed set because it does contain all of its limit points; no point is a limit point! A set that has no limit points is closed, by default, because it contains all of its limit points. Every intersection of closed sets is closed, and every finite union of closed sets is closed. 1.4 Topological Spaces We now consider a more general case of spaces without metrics, where we can still make sense of (or rather define appropriately) the notions of open and closed sets. These spaces are called topological spaces. Definition 1.4.1. A topological space is a pair (X, τ ) where X is a set and τ is a set of subsets of X satisfying certain axioms. τ is called a topology. Since this is not particularly enlightening, we must clarify what a topology is. Definition 1.4.2. A topology τ on a set X consists of subsets of X satisfying the following properties: 1. The empty set ∅ and the space X are both sets in the topology. 2. The union of any collection of sets in τ is contained in τ. 3. The intersection of any finitely many sets in τ is also contained in τ. 11 Mathematics 490 – Introduction to Topology Winter 2007 The following examples will help make this concept more clear. Example 1.4.3. Consider the following set consisting of 3 points; X = {a, b, c} and determine if the set τ = {∅, X, {a}, {b}} satisfies the requirements for a topology. This is, in fact, not a topology because the union of the two sets {a} and {b} is the set {a, b}, which is not in the set τ Example 1.4.4. Find all possible topologies on X = {a, b}. 1. ∅, {a, b} 2. ∅, {a}, {a, b} 3. ∅, {b}, {a, b} 4. ∅, {a}, {b}, {a, b} The reader can check that all of these are topologies by making sure they follow the 3 properties above. The first topology in the list is a common topology and is usually called the indiscrete topology; it contains the empty set and the whole space X. The following examples introduce some additional common topologies: Example 1.4.5. When X is a set and τ is a topology on X, we say that the sets in τ are open. Therefore, if X does have a metric (a notion of distance), then τ ={all open sets as defined with the ball above} is indeed a topology. We call this topology the Euclidean topology. It is also referred to as the usual or ordinary topology. Example 1.4.6. T If Y ⊆ X and τx is a topology on X, one can define the Induced topology as τy = {O Y |O ∈ τx }. This last example gives one reason why we must only take finitely many intersections when defining a topology. Example 1.4.7. Let X = R with the usual topology. Then certainly in this standard Eu- clidean T∞ topology, (−1/n, 1/n) is an open set for any integer n. However, the infinite intersection n=1 (−1/n, 1/n) is the set containing just 0. Thus, it is a single point set, which is not open in this topology. 1.5 Closed Sets (Revisited) As promised, we can now generalize our definition for a closed set to one in terms of open sets alone which removes the need for limit points and metrics. Definition 1.5.1. A set C is closed if X − C is open. Now that we have a new definition of a closed set, we can prove what used to be definition 1.3.3 as a theorem: A set C is a closed set if and only if it contains all of its limit points. 12 Mathematics 490 – Introduction to Topology Winter 2007 Proof. Suppose a set A is closed. If it has no limit points, there is nothing to check as it trivially contains its limit points. Now suppose z is a limit point of A. Then if z ∈ A, it contains this limit point. So suppose for the sake of contradiction that z is a limit point and z is not in A. Now we have assumed A was closed, so its complement is open. Since z is not in A, it is in the complement of A, which is open; which means there is an open set U containing z contained in the complement of A. This contradicts that z is a limit point because a limit point is, by definition, a point such that every open set about z meets A. Conversely, if A contains all its limit points, then its complement is open. Suppose x is in the complement of A. Then it can not be a limit point (by the assumption that A contains all of its limit points). So x is not a limit point which means we can find some open set around x that doesn’t meet A. This proves the complement is open, i.e. every point in the complement has an open set around it that avoids A. Example 1.5.2. Since we know the empty set is open, X must be closed. Example 1.5.3. Since we know that X is open, the empty set must be closed. Therefore, both the empty set and X and open and closed. 1.6 Continuity In topology a continuous function is often called a map. There are 2 different ideas we can use on the idea of continuous functions. Calculus Style Definition 1.6.1. f : Rn → Rm is continuous if for every  > 0 there exists δ > 0 such that when |x − x0 | < δ then |f (x) − f (x0 )| < . The map is continuos if for any small distance in the pre-image an equally small distance is apart in the image. That is to say the image does not “jump.” Topology Style In tpology it is necessary to generalize down the definition of continuity, because the notion of distance does not always exist or is different than our intuitive idea of distance. Definition 1.6.2. A function f : X → Y is continuous if and only if the pre-image of any open set in Y is open in X. If for whatever reason you prefer closed sets to open sets, you can use the following equivalent definition: Definition 1.6.3. A function f : X → Y is continuous if and only if the pre-image of any closed set in Y is closed in X. Let us give one more definition and then some simple consequences: 13 Mathematics 490 – Introduction to Topology Winter 2007 Definition 1.6.4. Given a point x of X, we call a subset N of X a neighborhood of X if we can find an open set O such that x ∈ O ⊆ N. 1. A function f : X → Y is continuous if for any neighborhood V of Y there is a neighborhood U of X such that f (U ) ⊆ V. 2. A composition of 2 continuous functions is continuous. Figure 1.1: Continuity of f with neighborhoods. 1.7 Homeomorphisms Homeomorphism is the notion of equality in topology and it is a somewhat relaxed notion of equality. For example, a classic example in topology suggests that a doughnut and coffee cup are indistinguishable to a topologist. This is because one of the geometric objects can be stretched and bent continuously from the other. The formal definition of homeomorphism is as follows. Definition 1.7.1. A homeomorphism is a function f : X → Y between two topological spaces X and Y that is a continuous bijection, and has a continuous inverse function f −1. 14 Mathematics 490 – Introduction to Topology Winter 2007 Figure 1.2: Homeomorphism between a doughnut and a coffee cup. Another equivalent definition of homeomorphism is as follows. Definition 1.7.2. Two topological spaces X and Y are said to be homeomorphic if there are continuous map f : X → Y and g : Y → X such that f ◦ g = IY and g ◦ f = IX. Moreover, the maps f and g are homeomorphisms and are inverses of each other, so we may write f −1 in place of g and g −1 in place of f. Here, IX and IY denote the identity maps. When inspecting the definition of homeomorphism, it is noted that the map is required to be continuous. This means that points that are “close together” (or within an  neighborhood, if a metric is used) in the first topological space are mapped to points that are also “close together” in the second topological space. Similar observation applies for points that are far apart. As a final note, the homeomorphism forms an equivalence relation on the class of all topo- logical spaces. Reflexivity: X is homeomorphic to X. Symmetry: If X is homeomorphic to Y , then Y is homeomorphic to X. Transitivity: If X is homeomorphic to Y , and Y is homeomorphic to Z, then X is home- omorphic to Z. The resulting equivalence classes are called homeomorphism classes. 15 Mathematics 490 – Introduction to Topology Winter 2007 1.8 Homeomorphism Examples Several examples of this important topological notion will now be given. Example 1.8.1. Any open interval of R is homeomorphic to any other open interval. Consider X = (−1, 1) and Y = (0, 5). Let f : X → Y be 5 f (x) = (x + 1). 2 Observe that f is bijective and continuous, being the compositions of addition and multiplication. Moreover, f −1 exists and is continuous: 2 f −1 (x) = x − 1. 5 Note that neither [0, 1] nor [0, 1) is homeomorphic to (0, 1) as such mapping between these intervals, if constructed, will fail to be a bijection due to endpoints. Example 1.8.2. There exists homeomorphism between a bounded and an unbounded set. Sup- pose 1 f (x) =. x Then it follows that (0, 1) and (1, ∞) are homeomorphic. It is interesting that we are able to “stretch” a set to infinite length. Example 1.8.3. Any open interval is, in fact, homeomorphic to the real line. Let X = (−1, 1) and Y = R. From the previous example it is clear that the general open set (a, b) is homeomor- phic to (−1, 1). Now define a continuous map f : (−1, 1) → R by  πx  f (x) = tan. 2 This continuous bijection possesses a continuous inverse f −1 : R → (−1, 1) by 2 f −1 (x) = arctan(x). π Hence f : (−1, 1) → R is a homeomorphism. Example 1.8.4. A topologist cannot tell the difference between a circle S 1 = {(x, y) ∈ R2 | x2 + y 2 = 1} and a square T = {(x, y) ∈ R2 | |x| + |y| = 1}, as there is a function f : S 1 → T defined by   x y f (x, y) = , |x| + |y| |x| + |y| which is continuous, bijective, and has a continuous inverse ! x y f −1 (x, y) = p ,p. x2 + y 2 x2 + y 2 Both circle and square are therefore topologically identical. S 1 and T are sometimes called simple closed curves (or Jordan curves). 16 Mathematics 490 – Introduction to Topology Winter 2007 2.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -2.5 Figure 1.3: The graph of f −1 (x). Example 1.8.5. S 1 with a point removed is homeomorphic with R. Without loss of generality, suppose we removed the North Pole. Then the stereographic projection is a homeomorphism between the real line and the remaining space of S 1 after a point is omitted. Place the circle “on” the x-axis with the point omitted being directly opposite the real line. More precisely, let S 1 = {(x, y) ∈ R | x2 +(y −1/2)2 = 1/4} and suppose the North Pole is N = (0, 1). Using geometry, we may construct f : S 1 \ N → R by defining 2x f (x, y) =. 1−y f is well-defined and continuous as the domain of f excludes y = 1, i.e. the North Pole. With the continuous inverse function 4x x2 − 4   −1 f (x) = , , x2 + 4 x2 + 4 we have f ◦ f −1 = f −1 ◦ f = I, hence f is a homeomorphism. Stereographic projection is a mapping that plays pivotal roles in cartography (mapmaking) and geometry. One interesting properties of stereographic projection is that this mapping is conformal, namely it preserves the angles at which curves cross each other. Example 1.8.6. The annulus A = {(x, y) ∈ R2 | 1 ≤ x2 + y 2 ≤ 4} is homeomorphic to the cylinder C = {(x, y, z) ∈ R3 | x2 + y 2 = 1, 0 ≤ z ≤ 1} since there exists continuous function 17 Mathematics 490 – Introduction to Topology Winter 2007 Figure 1.4: Stereographic projection of S 2 \ N to R2. f : C → A and g : A → C ! x y p f (x, y, z) = ((1 + z)x, (1 + z)y) and g(x, y) = p ,p , x2 + y 2 − 1 x2 + y 2 x2 + y 2 such that f ◦ g = g ◦ f = I. Thus f and g are homeomorphisms. 1.9 Theorems On Homeomorphism Theorem 1.9.1. If f : X → Y is a homeomorphism and g : Y → Z is another homeomorphism, then the composition g ◦ f : X → Z is also a homeomorphism. Proof. We need to establish that g ◦ f is continuous and has a continuous inverse for it to be a homeomorphism. The former is obvious as the composition of two continuous maps is continuous. Since f and g are homeomorphisms by hypothesis, there exists f −1 : Y → X and g −1 : Z → Y that are continuous. By the same token, f −1 ◦ g −1 is continuous and (g ◦ f ) ◦ f −1 ◦ g −1 = g ◦ f ◦ f −1 ◦ g −1 = g ◦ g −1 = IY.

Similarly (f ◦ g) ◦ g −1 ◦ f −1 = IX. Therefore g ◦ f is a homeomorphism.
18 Mathematics 490 – Introduction to Topology Winter 2007 Various topological invariants will be discussed in Chapter 3, including connectedness, com- pactness, and Hausdorffness. Recall that homeomorphic spaces share all topological invariants. Hence the following theorem (without proof) immediately follows: Theorem 1.9.2. Let X and Y be homeomorphic topological spaces. If X is connected or compact or Hausdorff, then so is Y. 1.10 Homeomorphisms Between Letters of Alphabet Letters of the alphabet are treated as one-dimensional objects. This means that the lines and curves making up the letters have zero thickness. Thus, we cannot deform or break a single line into multiple lines. Homeomorphisms between the letters in most cases can be extended to homeomorphisms between R2 and R2 that carry one letter into another, although this is not a requirement – they can simply be a homeomorphism between points on the letters. 1.10.1 Topological Invariants Before classifying the letters of the alphabet, it is helpful to get a feel for the topological invariants of one-dimensional objects in R2. The number of 3-vertices, 4-vertices, (n-vertices, in fact, for n ≥ 3), and the number of holes in the object are the topological invariants which we have identified. 1.10.2 Vertices The first topological invariant is the number and type of vertices in an object. We think of a vertex as a point where multiple curves intersect or join together. The number of intersecting curves determines the vertex type. Definition 1.10.1. An n-vertex in a subset L of a topological space S is an element v ∈ L such that there exists some neighborhood N0 ⊆ S of v where all neighborhoods N ⊆ N0 of v satisfy the following properties: 1. N ∩ L is connected. 2. The set formed by removing v from N ∩ L, i.e., {a ∈ N ∩ L | a 6= v}, is not connected, and is composed of exactly n disjoint sets, each of which is connected. The definition of connectedness will be discussed much more in the course and will not be given here. Intuitively, though, a set is connected if it is all in one piece. So the definition of an n-vertex given above means that when we get close enough to an n-vertex, it looks like one piece, and if we remove the vertex from that piece, then we get n separate pieces. We can say that for a given n ≥ 3, the number of n-vertices is a topological invariant because homeomorphisms preserve connectedness. Thus, the connected set around a vertex must map to another connected set, and the set of n disjoint, connected pieces must map to another set of n disjoint connected pieces. 19 Mathematics 490 – Introduction to Topology Winter 2007 The number of 2-vertices is not a useful topological invariant. This is true because every curve has an infinite number of 2-vertices (every point on the curve not intersecting another curve is a 2-vertex). This does not help us to distinguish between classes of letters. Figure 1.5: 3-vertex homeomorphism example Three curves intersecting in a 3-vertex are homeomorphic to any other three curves intersecting in a 3-vertex. However, they are not homeomorphic to a single curve. This point is illustrated in Figure 1.5. Similarly, any set close to a 4-vertex is homeomorphic to any other set close to a 4-vertex. In this case, “close” means a small neighborhood in which there are no other n-vertices (n ≥ 3). Figure 1.6 shows an example of this. Figure 1.6: 4-vertex homeomorphism example 1.10.3 Holes It is very clear when a one dimensional shape in two dimensional space has a hole: a shape possessing a hole closes in on itself. As a result, a topological structure with a hole cannot be continuously shrunk to a single point. Moreover, a function mapping a space with a hole to one without a hole cannot be a homeomorphism. To see this point, we note that the tearing of the circle means that nearby points on the circle can be mapped to very distant points on the line. This point, which is illustrated by figure 5, violates the continuity requirements of a homeomorphism. The number of holes is therefore an important topological invariant when classifying letters of the alphabet. 20 Mathematics 490 – Introduction to Topology Winter 2007 Figure 1.7: A loop is homeomorphic to any other loop (top), but not to a path with endpoints (bottom) 1.11 Classification of Letters We now use the preceeding arguments to classify the letters of the alphabet based on their topological invariants. We must use the letters printed as in the project specifications since changing the font can change the topological invariants of the letter. 21 Mathematics 490 – Introduction to Topology Winter 2007 0 holes, 0 three-vertices, 0 four-vertices: C, G, I, J, L, M, N, S, U, V, W, Z 1 hole, 0 three-vertices, 0 four-vertices: D, O 0 holes, 1 three-vertex, 0 four-vertices: E, F, T, Y 1 hole, 1 three-vertex, 0 four-vertices: P 0 holes, 2 three-vertices, 0 four-vertices: H, K 1 hole, 2 three-vertices, 0 four-vertices: A, R 2 holes, 2 three-vertices, 0 four-vertices: B 0 holes, 0 three-vertices, 1 four-vertex: X 1 hole, 0 three-vertices, 1 four-vertex: Q Table 1.1: Classification of the letters using the topological invariants None of the letters possess a n-vertex with n greater than five, so our classification scheme is based on the number of holes, the number of 3-vertices, and the number of 4-vertices in the letter. 1.11.1 The curious case of the “Q” It is interesting to consider the question of whether there exists a homeomorphism of R2 that carries Q homeomorphically to. Well, there is not! Intuitively, this is because such a function would need to map some points on the inside of the loop to points on the outside of the loop. This means that every point in the interior of the loop must map outside of the loop, since otherwise the image of one of the interior points would have to be on the loop in , by continuity, which would not be an invertible operation since that point in the image would have two points in the domain mapping to it. An analogous argument for the inverse function means that points on the outside of the loop in must all map to the inside of Q, which does not allow for there to be one segment inside and one segment outside. We can, however, define a homeomorphism f : Q → , where these symbols represent only the subsets of R2 included in each figure. It is defined simply by mapping q ∈ Q to the same point on the loop if q is on the loop, to the lower leg in if q is on the inner leg in Q, and to the upper leg in if q is on the outer leg in Q. The mapping on the legs would preserve the ratio of 22 Mathematics 490 – Introduction to Topology Winter 2007 Figure 1.8: Neighborhoods U and V. the distance along the leg from the 4-vertex versus the total length of the leg, so that it would be continuous at the 4-vertex. 1.12 Topological Invariants A topological invariant of a space X is a property that depends solely on the topology of the space X. That is, a property shared by any other space that is homeomorphic to X. Intuitively, a homeomorphism between X and Y maps points in X that are “close together” to points in Y that are “close together”, and points in X not “close together” to points in Y that are not “close together”. Below we will give a brief introduction to some of the topological invariants encountered during the course, and also give some simple examples to let the readers gain an insight on what these invariants are. (For more rigorous mathematical formulation, please refer to Chapter 3) 1.12.1 Hausdorff Property Definition 1.12.1. If X is a topological space and S is a subset of X then a neighbourhood of S is a set V, which contains an open set U containing S. i.e. S ⊆ U ⊆ V ⊆ X Definition 1.12.2. Let X be a topological space. Let x,y  X. We say that x and y can be separated by neighbourhoods if there exists T a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint i.e. U V = ∅. Definition 1.12.3. A space is Hausdorff if every two distinct points have disjoint neighbour- hoods. Example 1.12.4. A topological space is called T1 if for any pairs of point x,y ∈ X, there is an open set Ox such that x ∈ Ox and y isn’t. Hausdorff are T1 but the vice versa is not always true. A simple example is an infinite set endowed with the cofinite topology. A nice property of Hausdorff spaces is that compact sets are always closed. This may fail for spaces which are non-Hausdorff. The definition of a Hausdorff space says that points can be sep- arated by neighborhoods. It turns out that this implies something which is seemingly stronger: 23 Mathematics 490 – Introduction to Topology Winter 2007 in a Hausdorff space every pair of disjoint compact sets can be separated by neighborhoods. This is an example of the general rule that compact sets often behave like points. 1.12.2 Compactness Property Definition 1.12.5. A space is compact if every open cover has a finite subcover. (This is equivalent to “closed” and “bounded” in an Euclidean Space.) One of the main reasons for studying compact spaces is because they are in some ways very similar to finite sets: there are many results which are easy to show for finite sets, whose proofs carry over with minimal change to compact spaces. Example 1.12.6. Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. Then we can separate x and A by neighbourhoods: for each a in A, let U(x) and V(a) be disjoint neighbourhoods containing x and a, respectively. Then the intersection of all the U(x) and the union of all the V(a) are the required neighbourhoods of x and A. Note that if A is infinite, the proof fails, because the intersection of arbitrarily many neigh- bourhoods of x might not be a neighbourhood of x. The proof can be ”rescued”, however, if A is compact: we simply take a finite subcover of the cover V (a) of A. In this way, we see that in a Hausdorff space, any point can be separated by neighbourhoods from any compact set not containing it. In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighbourhoods. Here we give some theorems related to compactness: Theorem 1.12.7. If f is a continuous function, then the image of a compact set under f is also compact. Note the above theorem implies that compactness is a topological invariant. Theorem 1.12.8. Any closed subset C of a compact space K is also compact. Given a non-compact set X, sometimes one would like to construct a homeomorphism to an open dense set in a compact space X. e Intuitively, it is like adding some points to X to make it compact. Theorem 1.12.9. For any non-compact space X the one-point compactification of X is obtained by adding one extra point ∞ and defining the open sets of the new space to be the open sets of X together with the sets of the form G ∪ ∞, where G is an open subset of X such that X \ G is compact. Example 1.12.10. Any locally compact (every point is contained in a compact neighbourhood) Hausdorff space can be turned into a compact space by adding a single point to it, by means of one-point compactification. The one-point compactification of R is homeomorphic to the circle S 1 ; the one-point compactification of R2 is homeomorphic to the sphere S 2. Using the one- point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space. 24 Mathematics 490 – Introduction to Topology Winter 2007 1.12.3 Connectedness and Path Connectedness Properties Definition 1.12.11. A space X is connected if it is not the union of a pair of disjoint non- empty open sets. Equivalently, a space is connected if the only sets that are simultaneously open and closes are the whole space and the empty set. Another equivalent formal definition is: Definition 1.12.12. A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. Example 1.12.13. The closed interval [0, 2] is connected; it can, for example, be written as the union of [0, 1) and [1, 2], but the second set is not open in the topology of [0, 2]. On the other hand, the union of [0, 1) and (1, 2] is disconnected; both of these intervals are open in the topological space [0, 1) ∪ (1, 2]. A major theorem of connectedness is: Theorem 1.12.14. Let X and Y be topological space. If f: X 7→ Y is a continuous function, then the image of a connected subset C  X under f is also connected. Note the above theorem implies that connectedness is a topological invariant. It can also be considered a generalization of the intermediate value theorem. Connectedness is one of the principal topological properties that is used to distinguish topolog- ical spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path. Definition 1.12.15. The space X is said to be path-connected if for any two points x,y  X there exists a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. (This function is called a path from x to y.) Theorem 1.12.16. A path-connected space is connected. Note the vice-versa of the above theorem is not always true. Example 1.12.17. Consider the space defined as the graph of the function sin(1/x) over the interval (0, 1] extended by the single point (0,0). This set is then equipped with the topology induced from the Euclidean plane. It is connected but not path-connected. It is the continuous image of a locally compact space (namely, let V be the space -1 ∪ (0, 1], and use the map f from V to T defined by f(-1) = (0, 0) and f(x) = (x, sin(1/x))), but is not locally compact itself. 25 Mathematics 490 – Introduction to Topology Winter 2007 ! 1 Figure 1.9: Image of the curve sin. x 26 Chapter 2 Making New Spaces From Old In Chapter 1 we learned the technical definition of the topological spaces - which are in some sense generalizations of metric spaces - that will be the object of study for the rest of these notes. Moreover, we saw how to construct and interpret functions f : X → Y that map points in one topological space X to points in another space Y. These maps, which may or may not be continuous, are therefore relationships between spaces and it is typically impossible to say how the topological properties of the domain space X and the image space Y are related. When these spaces are deformed continuously via a homeomorphism, however, such properties are topological invariants. This important result gives an approach towards determining whether or not spaces are topologically “different”. We now move on to describing methods that can be used to create new topological spaces that we will study further in future chapters. Interesting connections between old and familiar spaces will also be drawn in this chapter. As we shall see, this subject is interesting and valuable for a number of reasons. For example, there simply is aesthetic beauty in building up complex structures and spaces from simple ones. But on a more fundamental level, many properties of the initial and final spaces are related when these techniques are applied. In particular, we consider product spaces, quotient spaces and group actions, and identification spaces. 2.1 Cartesian Products of Spaces Let us forget about topology for a moment, take a step backwards, and consider the function f (x, y) = xy. When we look at this function, we typically think of its behavior as the variables x and y are varied separately. That is, for a fixed value of x we can observe the change in the function’s value as we vary y, and vice-versa. In this sense we naturally “factor” the function f (x, y). We can apply similar intuition to certain topological spaces that can be “factored” as the product X × Y. Here, points in the product space X × Y are represented by Cartesian pairs (x, y) where x ∈ X and y ∈ Y. The spaces X and Y are typically “simpler”, more familiar, or more easily visualized spaces, though this is not necessarily the case. There exists ample examples of decomposing familiar topological spaces as product spaces. The plane R2 , for example, can be written as the product of two copies of the real line R or R × R. 27 Mathematics 490 – Introduction to Topology Winter 2007 The definition of Cartesian products certainly agrees with our picture of representing points as having an x-coordinate (from the first R in R × R) and a y-coordinate (from the second R in R × R). Also, the surface of a cylinder can be thought of as a circle extruded down the length of an interval. We can therefore represent the cylinder as Cyl = S 1 × I, where S 1 is a circle in the (x,y) plane and I is an interval along the z axis. Likewise, the torus can be interpreted as one small circle extruded around the circumference of a second larger circle and hence written as T = S 1 × S 1. Rather exotic spaces can also be built out of familiar ones by taking products. Ultimately there are an infinitely large number of ways to combine such spaces as the disk, the sphere, the torus, and the real projective plane, but examples include the spaces RP 2 × RP 2 × T and RP 2 ×... × RP 2. While these may have little meaning to us as we cannot picture them in our minds or draw them on paper, they nevertheless represent real objects that can be considered. 2.2 The Product Topology We are therefore armed with a mechanism for “combining” spaces using products. Our next step must therefore be to extend this picture to include open sets and topology. In order to make this intellectual leap, we can observe that if U and V are open sets in the component spaces X and Y, respectively, then, if the world is at all fair, U × V should be an open set in the product X × Y. Making this a requirement means that, just as we can build-up arbitrary open sets in X adding open sets in a base for its topology, we can likewise build-up arbitrary open sets in X × Y as the union of open sets of the form U × V. The collection of such sets U × V can be called the family B, and the finite intersection of members in B also lies in B. The collection B is therefore a base for a topology on X × Y that is appropriately named the product topology. When the set X × Y is paired with this topology, it is referred to as a product space. Now that we have a notion of the product topology, it is possible to consider the relationship between open sets in a product space and those in the component spaces. As such, we define the projections pX : X × Y → X with pX (x, y) = x and pY : X × Y → Y with pY (x, y) = y. This allows us to describe the product topology in terms of the projections. Theorem 2.2.1. When X × Y is equipped with the product topology, the projections pX and pY are continuous functions and are open maps. In addition, the product topology on X × Y is the smallest topology for which the projections are continuous. This important result is true because open boxes U × V in the product space are, by definition, mapped to open sets U and V in the component spaces, respectively. In a reciprocal fashion, it is clear that the pre-image of open sets U , V in the component spaces are open strips U × Y , V × X in the product space. Moreover, any given topology on a product space must necessarily contain the open sets U × V and is therefore at least as large as the product topology. Another general result is that we can check that the continuity of a function f mapping compo- nent spaces to product spaces by composing it with the projections and checking the continuity of the resulting functions. Theorem 2.2.2. The function f : Z → X × Y is continuous if and only if the composite functions pX f : Z → X and pY f : Z → Y are continuous. 28 Mathematics 490 – Introduction to Topology Winter 2007 Figure 2.1: As discussed in the notes, the cylinder (left) and the torus (right) can be represented as product spaces. Notice the graphical relationship between open sets in the component spaces X and Y and those in X × Y , which is crucial for developing the product topology. These notions of product spaces and topology can be made more concrete by comparing these definitions with figure one and its examples. 2.3 Properties of Product Spaces There are also a number of nice theorems that allow us to relate the topological properties of product spaces X × Y to those of the component spaces X and Y. While these theorems will be discussed in detail later in these notes, we nevertheless state them here. Theorem 2.3.1. The product space X × Y is a Hausdorff space if and only if both X and Y are Hausdorff. Theorem 2.3.2. The product space X × Y is compact if and only if both X and Y are compact. Theorem 2.3.3. If X and Y are connected spaces then the product space X×Y is also connected. It is also clear that we can extend the results shown below to products of any number of such as spaces, as in the case of X1 × X2 ×... × Xn. This is true since we can simply make the substitutions X 7→ X1 and Y 7→ X2 ×... × Xn in these theorems and make an inductive argument. 29 Mathematics 490 – Introduction to Topology Winter 2007 2.4 Identification Spaces Before introducing Identification spaces, let’s first define a partition. Let XS be a topological space and let P be a family of disjoint nonempty subsets of X such that P = X. Such a family is usually called a partition of X. That is, partitions of X are disjoint subsets of X S that together form X. For a trivial example, let elements of P be the points of X so that X = P. We may form a new space Y , called an identification space, as follows. The points of Y are the members of P and, if π : X → Y sends each point of X to the subset of P containing it, the topology of Y is the largest for which π is continuous. Therefore, a subset O of Y is open if and only if π −1 (O) is open in X. This topology is called the identification topology on Y. (We call Y an identification space, because all points in X that are sent to the same subset of Y have become the same point.) Example 2.4.1. Cylinder.   [ [ h[ i Y = {(a, b)} {(0, b), (1, b)}. a6=0,1 0≤b≤1 Example 2.4.2. Möbius band. Example 2.4.3. Torus.   " # " #  [ [ [ [ [ [ Y =  {(a, b)}   {(0, b), (1, b)} {(a, 0), (a, 1)} [{(0, 0), (0, 1), (1, 0), (1, 1)}]. a6=0,1 0