Part I A Geometric Introduction to Topology “chap01” — 2003/2/19 — page1—#1 “chap01” — 2003/2/19 — page2—#2 1 Basic point set topology 1.1 Topology in Rn Topology is the branch of geometry that studies “geometrical objects” under the equivalence relation of homeomorphism. A homeomorphism is a function f : X → Y which is a bijection (so it has an inverse f −1 : Y → X) with both f and f −1 being continuous. One of the prime aims of this chapter will be to enhance our understanding of the concept of continuity and the equivalence relation of homeomorphism. We will also discuss more precisely the “geometrical objects” in which we are interested (called topological spaces), but our viewpoint will primarily be to understand more familiar spaces better (such as surfaces) rather than to explore the full generalities of topological spaces. In fact, all of the spaces we will be interested in exist as subspaces of some Euclidean space Rn. Thus our first priority will be to understand continuity and homeomorphism for maps f : X → Y , where X ⊂ Rn and Y ⊂ Rm. We will use bold face x to denote points in Rk. One of the methods of mathematics is to abstract central ideas from many examples and then study the abstract concept by itself. Although it often seems to the student that such an abstraction is hard to relate to in that we are fre- quently disregarding important information of the particular examples we have in mind, the technique has been very successful in mathematics. Frequently, the success is rooted in the following idea: knowing less about something limits the avenues of approach available in studying it and this makes it easier to prove theorems (if they are true). Of course, the measure of the success of the abstrac- ted idea and the definitions it suggests is frequently whether the facts we can prove are useful back in the specific situations which led us to abstract the idea in the first place. Some of the most important contributions to mathematics have been made by those who have figured out good definitions. This is difficult for the student to appreciate since definitions are usually presented as if they came from some supreme being. It is more likely that they have evolved through many wrong guesses and that what is presented is what has survived the test of time. 3 “chap01” — 2003/2/19 — page3—#3 4 1. Basic point set topology It is also quite possible that definitions and concepts which seem so right now (or at least after a lot of study) will end up being modified at a later stage. We now recall from calculus the definition of continuity for a function f : X → Y , where X and Y are subsets of Euclidean spaces. Definition 1.1.1. f is continuous at x ∈ X if, given >0, there is a δ>0so that d(x, y) <δimplies that d(f(x),f(y)) <. Here d indicates the Euclidean 2 2 1/2 distance function d((x1,...,xk), (y1,...,yk))=((x1 −y1) +···+(xk −yk) )) . We say that f is continuous if it is continuous at x for all x ∈ X. It will be convenient to have a slight reformulation of this definition. For z ∈ Rk, we define the ball of radius r about z to be the set B(z,r)={y ∈ Rk : d(z, y) <r} If C is a subset of Rk and z ∈ C, then we will frequently be interested in the intersection C ∩B(z,r), which just consists of those points of C which are within distance r of z. We denote by BC (x,r)=C ∩ B(x, r)={y ∈ C : d(y, x) <r}. Our reformulation is given in the following definition. Definition 1.1.2. f : X → Y is continuous at x ∈ X if given >0, there is a −1 δ>0 so that BX (x,δ) ⊂ f (BY (f(x),)). f is continuous if it is continuous at x for all x ∈ X. Exercise 1.1.1. Show that the reformulation Definition 1.1.2 is equivalent to the original Definition 1.1.1. This requires showing that, if f is continuous in Definition 1.1.1, then it is also continuous in Definition 1.1.2, and vice versa. We reformulate in words what Definition 1.1.2 requires. It says that a function is continuous at x if, when we look at the set of points in X that are sent to a ball of radius about f(x), no matter what >0 is given to us, then this set always contains the intersection of a ball of some radius δ>0 about x with X. This definition leads naturally to the concept of an open set. Definition 1.1.3. A set U ⊂ Rk is open if given any y ∈ U, then there is a number r>0 so that B(y,r) ⊂ U.IfX is a subset of Rk and U ⊂ X, then we say that U is open in X if given y ∈ U, then there is a number r>0 so that BX (y,r) ⊂ U. In other words, U is an open set in X if it contains all of the points in X that are close enough to any one of its points. What our second definition is saying −1 in terms of open sets is that f (BY (y,)) satisfies the definition of an open set in X containing x; that is, all of the points in X close enough to x are in it. Before we reformulate the definition of continuity entirely in terms of open sets, we look at a few examples of open sets. Example 1.1.1. Rn is an open set in Rn. Here there is little to check, for given x ∈ Rn, we just note that B(x,r) ⊂ Rn, no matter what r>0 is. Example 1.1.2. Note that a ball B(x,r) ⊂ Rn is open in Rn.Ify ∈ B(x,r), then if r = r − d(y, x), then B(y,r) ⊂ B(x,r). To see this, we use the triangle inequality for the distance function: d(z, y) <r implies that d(z, x) ≤ d(z, y)+d(y, x) <r + d(y, x)=r. Figure 1.1 illustrates this for the plane. “chap01” — 2003/2/19 — page4—#4 1.1. Topology in Rn 5 y z x Figure 1.1. Balls are open. Figure 1.2. Open and closed rectangles. Example 1.1.3. The inside of a rectangle R ⊂ R2, given by a<x<b,c< y<d, is open. Suppose (x, y) is a point inside of R. Then let r = min(b − x, x − a, d − y, y − c). Then if (u, v) ∈ B((x, y),r), we have |u − x| <r,|v − y| <r, which implies that a<u<b,c<v<d,so(u, v) ∈ R. However, if the perimeter is included, the rectangle with perimeter is no longer open. For if we take any point on the perimeter, then any ball about the point will contain some point outside the rectangle. We illustrate this in Figure 1.2. Example 1.1.4. The right half plane, consisting of those points in the plane with first coordinate positive, is open. For given such a point (x, y) with x>0, then if r = x, the ball of radius r about (x, y) is still contained in the right half plane. For any (u, v) ∈ B((x, y),r) satisfies |u − x| <rand so x − u<x, which implies u>0. Example 1.1.5. An interval (a, b) in the line, considered as a subset of the plane (lying on the x-axis), is not open. Any ball about a point in it would have to contain some point with positive y-coordinate, so it would not be contained in (a, b). Note, however, that it is open in the line, because, if x ∈ (a, b) and r = min(b − x, x − a), then the intersection of the ball of radius r about x with the line is contained in (a, b). Of course, the line itself is not open in the plane. Thus we have to be careful in dealing with the concept of being open in X, where “chap01” — 2003/2/19 — page5—#5 6 1. Basic point set topology X is some subset of a Euclidean space, since a set which is open in X need not be open in the whole space. Exercise 1.1.2. Determine whether the following subsets of the plane are open. Justify your answers. (a) A = {(x, y):x ≥ 0}, (b) B = {(x, y):x =0}, (c) C = {(x, y):x>0 and y<5}, (d) D = {(x, y):xy < 1 and x ≥ 0}, (e) E = {(x, y):0≤ x<5}. Note that all of these sets are contained in A. Which ones are open in A? We now give another reformulation for what it means for a function to be continuous in terms of the concept of an open set. This is the definition that has proved to be most useful to topology. Definition 1.1.4. f : X → Y is continuous if the inverse image of an open set in Y is an open set in X. Symbolically, if U is an open set in Y , then f −1(U)is an open set in X. Note that this definition is not local (i.e. it is not defining continuity at one point) but is global (defining continuity of the whole function).