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Home , Homeomorphism

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Lecture 4: September 8 Homeomorphisms. As I mentioned in the first lecture, the purpose of is to look at qualitative properties of geometric objects that do not depend on the exact of an object, but more on how the object is put together. We formalized this idea by definiting topological spaces; but what does it mean to say that two di↵erent topological spaces (such as a and a square) are really “the same”? Definition 4.1. Let f : X Y be a bijective between topological spaces. ! 1 If both f and the f : Y X are continuous, then f is called a homeomorphism, and X and Y are said to! be homeomorphic. Intuitively, think of X as being made from some elastic material (like a balloon), and think of stretching, bending, or otherwise changing the shape of X without tearing the material. Any Y that you get in this way will be homeomorphic to the original X. Note that the actual definition is both more precise and more general, since we are allowing arbitrary functions. Suppose that f : X Y is a homeomorphism. For each U X,we ! 1 ✓ are assuming that its inverse image under f : Y X is open in X; but because f is bijective, this is the same as the image of U! under f. In other words, a homeomorphism is a bijective function f : X Y such that f(U) is open if and only if U is open. We therefore get a bijective! correspondence not only between the points of X and Y , but also between the open sets in both . So any question about the topology of X or Y will have the same answer on both sides; we may therefore think of X and Y as being essentially the same topological . Example 4.2. The real numbers R are homemorphic to the open (0, 1). One possible choice of homeomorphism is the function ex f : R (0, 1),f(x)= ! ex +1 1 Both f and the inverse function f (y) = log(y) log(1 y) are continuous. Example 4.3. Consider the function 1 f :[0, 1) S ,f(t) = (cos t, sin t) ! that takes the interval (with the from R)totheunitcircle (with the subspace topology from R2). It is bijective and continuous, but not a homeomorphism: [0, 1/2) is open in [0, 1), but its image is not open in S1. Example 4.4. Let us classify the letters of the English alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ up to homeomorphism. Here we think of each letter as being made from line segments in R2; the topology is the subspace topology. By inspection, there are eight homeomorphism classes, depending on the number of loops and line segments in each letter: BARPQDOCGIJLMNSUVWZEFTYHKX For example, W can be bent to make I, and so the two are homeomorphic. On the other hand, there is no homeomorphism between T and I: if we remove the crossing point, we are left with three intervals in the case of T, but removing one point from I produces at most two intervals. (Think about how one can say this in terms of the topology on each letter.) 2

Topological . In the remainder of today’s class, I want to introduce three additional examples of topological spaces. The first one is topological mani- folds. A is a space X that “locally” looks like : if you sit at any point of X, and only look at points nearby, you may think that you are in Rn. Here is the precise definition. Definition 4.5. An n-dimensional is a Hausdor↵ X with the following property: every point x X has a neighborhood that 2 is homeomorphic to an open subset in Rn. In , people look at other classes of manifolds that are obtained by working with a smaller class of functions. For example, if a function and its inverse function are both di↵erentiable, it is called a di↵eomorphism; di↵erentiable man- ifolds are defined by replacing “homeomorphic” by “di↵eomorphic” in the above definition. In algebraic geometry, there is a similar definition with polynomials. At this point, somebody asked why we need the Hausdor↵condition; the answer is that we do not want to allow something like taking two copies of R and gluing them together along R 0 . (More about this example later on, when we discuss \{ } quotient spaces.) Later in the semester, we will show that an open subset in Rn can never be homeomorphic to an open subset in Rm for m = n; this means that the of a manifold really is a well-defined notion. 6 Example 4.6. The square and the circle are both one-dimensional manifolds; a homeomorphism between them is given by drawing the square inside the circle and projecting one to the other from their common center. Example 4.7. The n- n n+1 2 2 2 S = (x0,x1,...,xn) R x + x + + x =1 , 2 0 1 ··· n with the subspace topology coming from Rn+1 , is an n-dimensional manifold. Intu- itively, this is clear; let me prove it for n = 2 by using . The z = 1 is tangent to the sphere at the south pole; given any point (x, y, z) not equal to the north pole (0, 0, 1), we can see where the line connecting (0, 0, 1) and (x, y, z) intersects the plane z = 1. In this way, we get a 2 2 f : S (0, 0, 1) R . \{ }! It is easy to work out the formulas to see that f and its inverse are continuous. The points on the line are parametrized by (0, 0, 1) + t(x, y, z 1), with t R;the intersection point with the plane has 2 2 1+t(z 1) = 1 or t = , 1 z which means that 2x 2y f(x, y, z)= , . 1 z 1 z 1 One can show in a similar manner that f is continuous. Since we can also do stereographic projection from the south pole, every point of S2 has a neighborhood 2 that is homeomorphic to R . Å ã Example 4.8. The from analysis gives us one way to define manifolds. Suppose that f : R2 R is a continuously di↵erentiable function. The ! inverse function theorem says that if f(x0,y0) = 0, and if the partial derivative 3

@f/@y does not vanish at the point (x0,y0), then all nearby solutions of the equation f(x, y) = 0 are of the form y = '(x) for a continously di↵erentiable function ':(x0 ", x0 + ") R with '(x0)=y0. This function ' gives us a homeomorphism between a small! neighborhood of the 1 1 point (x0,y0)inthesetf (0) and an open interval in R. This shows that f (0) is a one-dimensional manifold, provided that at least one of the two partial derivatives 1 @f/@x or @f/@y is nonzero at every point of f (0).

Example 4.9. If M1 and M2 are manifolds of dimension n1 and n2,respectively, then their product M1 M2 (with the ) is a manifold of dimension ⇥ 1 1 n1 + n2. The proof is left as an exercise. For instance, the product S S is a two-dimensional manifold called the . ⇥ An important general problem is to classify manifolds (or more general topolog- ical spaces) up to homeomorphism. In general, this is only possible if we impose suciently many other conditions (such as or compactness) to limit the class of topological spaces we are looking at. We will come back to this problem later in the semester.

Quotient spaces and the quotient topology. In geometry, it is common to describe spaces by “cut-and-paste” constructions like the following. Example 4.10. If we start from the unit square and paste opposite edges (with the same orientation), we get the torus. If we start from the closed unit in R2 and collapse the entire boundary into a point, we obtain S2. To make a M¨obius band, we take a strip of paper, twist one end by 180, and then glue the two ends together. We can make a torus with two holes by taking two copies of the torus, removing a small disk from each, and then pasting them together along the two boundary . In each of these cases, the result should again be a topological space. To formalize this type of construction, we start with a topological space X and an on it; intuitively, tells us which points of X should be glued together. (Recall that⇠ an equivalence⇠ relation is the same thing as a partition of X into disjoint subsets, namely the equivalence classes; two points are equivalent if and only if they are in the same .) What we want to do is to build a new topological space in which each equivalence class becomes just one point. To do this, we let X/ be the set of equivalence classes; there is an obvious function ⇠ p: X X/ , ! ⇠ which takes a point x X to the equivalence class containing x. Now there is a natural way to make X/2 into a topological space. ⇠ Proposition 4.11. The collection of sets 1 T = U X/ p (U) is open in X ✓ ⇠ defines a topology on X/ ,calledthequotient topology. ⇠ Proof. We have to check that the three conditions in the definition of topology are 1 1 satisfied. First, p ( )= and p (X/ )=X, and so both and X/ belong ; ; ⇠ ; ⇠ 4 to T . The conditions about unions and intersections follow from the set-theoretic formulas

1 1 1 1 1 p U = p (U ) and p (U V )=p (U) p (V ) i i \ \ i I ! i I [2 [2 and the definition of T . ⇤ With this definition, p becomes a . In fact, the quotient topology is the largest topology with the property that p is continous. Even when X is Hausdor↵, the quotient X/ is not necessarily Hausdor↵. ⇠ Example 4.12. Let us go back to the example of the line with two origins, made by gluing together two copies of R along R 0 . Here we can take X = 0, 1 R, and define the equivalence relation so that\{ (0} ,t) (1,t) for every t ={ 0.} Most⇥ equivalence classes have two points, namely (0,t⇠), (1,t) with t = 0,6 except for (0, 0) and (1, 0) . The quotient space X/{ is not Hausdor} ↵(because6 the two equivalence{ } classes{ } (0, 0) and (1, 0) cannot⇠ be separated by open sets), but { } { } every point has a neighborhood homeomorphic to R. In fact, it is an interesting problem of finding conditions on X and that will guarantee that X/ is Hausdor↵. This does happen in real life: I work in⇠ algebraic geometry, but in one⇠ of my papers, I had to spend about a page on proving that a certain quotient space was again Hausdor↵. The most useful property of the quotient topology is the following. Theorem 4.13. Let f : X Y be a continous function that is constant on equiv- alence classes: whenever x! x ,onehasf(x )=f(x ). Then f induces a 1 ⇠ 2 1 2 function f˜: X/ Y , which is continuous for the quotient topology on X/ .IfY is Hausdor↵, then⇠!X/ is also Hausdor↵. ⇠ ⇠ Proof. The proof is left as an exercise. ⇤ Product spaces and the product topology. We have already seen that the product of two topological spaces is again a topological space. Now we want to deal with the general case where we allow an arbitrary (and possibly infinite) number of factors. So let (Xi, Ti) be a collection of topological spaces, indexed by a (possibly infinite) set I. Consider the cartesian product

X = Xi = (xi)i I xi Xi for every i I , 2 2 2 i I Y2 whose elements are all (generally infinite) families of elements xi Xi, one for each i I. It is not completely obvious that X has any elements at2 all – at least, this does2 not follow from the usual axioms of set theory. In addition to the Zermelo- Frenkel axioms, one needs the so-called axiom of choice, which says that if Xi = for every i I,thenX = . 6 ; 2 6 ; Note. The axiom of choice claims that one can simultaneously choose one element from each of a possibly infinite number of nonempty sets. The problem is that we cannot just “choose” the elements arbitrarily, because we do not have enough time to make infinitely many choices. This axiom may seem very natural, but it has a large number of strange consequences. For example, you may have heard of the Banach-Tarski paradox: the axiom of choice implies that one can divide the three- dimensional unit into finitely many pieces, and then put them back together 5 in a di↵erent way and end up with a ball of twice the radius. This kind of thing lead to many arguments about the validity of the axiom, until it was proved that the axiom of choice is logically independent from the other axioms of set theory. Nowadays, most people assume the axiom of choice since it makes it easier to prove interesting theorems. We want to make X into a topological space. There are two di↵erent ways of generalizing the definition from the case of two factors. Definition 4.14. The box topology on X is the topology generated by the basis

U U T for every i I . i i 2 i 2 i I Y2 It is not hard to see that this is indeed a basis for a topology: it contains X, and since ß ™ U V = U V , i \ i i \ i i I ! i I ! i I Y2 Y2 Y2 the intersection of any two basic open sets is again a basic open set. It it clear from the definition that the coordinate functions

pj : X Xj,pj ((xi)i I )=xi ! 2 are continuous functions. The box topology is a perfectly good topology on X,but when I is infinite, it has a very large number of open sets, which leads to certain pathologies. (For example, it usually does not satisfy the first or second countability axiom.) We can get a better topology by putting some finiteness into the definition. Definition 4.15. The product topology on X is the topology generated by the basis

U U T for every i I, and U = X for all but finitely many i I . i i 2 i 2 i i 2 i I Y2 The di↵erence with the box topology is that we are now allowed to specify only finitely many coordinates in each basic open set. The idea behind the product topologyß is that every set of the form pj(U) should be open (since we want pj™to be continuous), and that finite intersections of open sets need to be open (since we want to have a topology). In fact, one can show that the product topology is the smallest topology on X that makes all the coordinate functions pj : X Xj continous. ! Theorem 4.16. If we give X the product topology, then a function f : Y X is continuous if and only if f = p f : Y X is continous for every i I. ! i i ! i 2 Proof. The proof is the same as in the case of two factors. ⇤ This nice result fails for the box topology. For that reason, we almost always use the product topology when talking about infinite products of topological spaces.

Example 4.17. Let X be the product of countably many copies of R,indexedby the set 1, 2,... . The function { } f : R X, f(t)=(t, t, . . . ) ! 6 is not continuous for the box topology, because

1 1 1 1 f , = 0 n n { } n=1 ! Y is not open in R.

Å ã Next time, we will talk about connectedness, 23 to 25 in the textbook. § §