Welcome back. Welcome to Lecture 5, in our course Analysis of a Complex Kind. Today, we'll talk a little bit about topology in the plane. Topology is the study of shapes, and we need to name some shapes and find out how to write them down in complex notation, and learn about some concepts of topology in order to move to some really neat applications next week. So let's start with something very simple. Circles in the complex plane. A circle needs a center and a radius. So let's call the center z0 equals x0 plus iy0. So suppose we wanted to write down a circle of center, z0 of radius r. The inside of that circle, we'll call that a disk. So this inside here is the disc of radius r, centered at z0, and the circle itself, we'll call that circle set the boundary. So by Br of z0, we denote the inside disc of radius r, centered at z0, and Kr is the actual circle. So z has distance less than r from z0 or z has distance r from z0, that's a long to write. So we need better way to write down what the distance is between z and z0. So how do we measure distance in the complex plane? Let's draw a picture. Suppose, I have a point z0, with coordinates x0 plus iy0, and another point z with coordinates x plus iy, and I wanted to find the distance between these two points, in particular, I want to determine whether this distance is at most r or equal to r. So what do I know? I know this is x0, and this is the x-coordinate x. Similarly, I know this is y0, and this is y. What is the distance between z and z0? It's the length of the connecting line segment between z0 and z. How do I find that? Well, I can draw a right triangle here, and then I find that triangle with this right angle. So in order to find this distance, I'll call that d, I could just as well find the length of the two lengths and use the Pythagorean theorem. So how long are these two legs? Well, this leg down here has length x minus x0, because that's the distance between the two x-coordinates. This leg here has length y minus y0. So in order to find the distance d, I can use the Pythagorean theorem and find that d is equal to the square root of the sum of the squares of the two lengths. So the square of x minus x0, and the square of y minus y0. But that looks awfully like the modulus of a complex number, namely, the complex number x minus x0, plus i times y minus y0. That's how we would calculate its modulus, and this complex number is really the complex number z minus z0. So this distance d, is actually the modulus of the complex number, z minus z0. With that new notation, I can write down the disc of radius r, centered at c0 in a much easier way. It's the set of all those zs, for which the modulus of z minus z0 is less than r, and the circle of radius r, centered at c0 is the set of all zs for which the modulus of z minus z0 is equal to r. We need some more definitions. Suppose we are given a set E, so some set. Maybe it looks like this, maybe it has a hole, who knows? We say an interior point of E is a point like this, point z0, such that there exists some r, such that the disc of radius r centered at z0 is entirely contained in E. So I can draw a little disk around this point, it still fits entirely into the set E. So obviously, if I tried to do that out here, so if I tried this point, like no matter how small a disk I draw, it's never going to be contained in E. But even a point like this, if I draw a disk around it, some part of that disk will always stick out. We're a point right here, on that edge. If it's right on the edge, no matter how small I draw the disc, it'll always stick out to it. Interior points are points that are really inside of E, and there's some room, some buffer room before I really end the end of E. So those are interior points of E. Points like this are called boundary points of E. So at point B is on the boundary of E. In every disk that I draw around it contain some points in E, restricted good ones. So I didn't need some points in E, but also some points that are not in E. So no matter how small I choose this disk, I will always capture some points in E, and some outside of E. Those points are called boundary points of E, and the boundary is denoted with this funny Nu Greek symbol that we have not introduced before. It's like a Delta symbol on this. So that's the symbol for the boundary of E. We say that a set is open if every one of its points is an interior point. On the other hand, we say it's closed if it contains all of its boundary points. So those are actually quite subtle definition. So we need to look at some examples to understand those a little bit better. So the ball of radius r centered at z0 is an example of an open set. Let's look at that. A disk centered at a points is z0 and a radius r. I'm claiming that's an open set. So no matter which point you pick, you always find a little disk around that point. It still fits into the original big disk, even if it's really close to the edge. I can't be on the edge because points on the edge are not included in my disc of radius r because I require the distance between z and z0 to be strictly less than r. So no matter where I am, I'm not quite on the boundary, and so there is always room for a little disk around it without hitting the boundary. Also, everything out here is also open, all those points for which z minus z0 is bigger than r, it's also open because I can find, for any point the little disk around it, that stays entirely outside of the disk of radius r. The whole plane is open. Well, every point is interior point in the whole plane, because no matter where I put a point in the plane, we can find the disk around that, it fits into the complex plane. So that's trivial. The empty set being open is a little funky, but the empty set doesn't have any interior points, and so it contains all of those. So that's a little bit of a mathematical technicality, that the empty set is also open. Now, here's an example of a closed set.The disk of radius r, but this time instead of writing a strict inequality, I made it a less than or equal to, so I didn't include these boundary points. Now, that time now it is a closed set because it's the disk together with all of its boundary points. So this includes all of its boundary points, and it's a closed set. The circle itself is also closed. There's no extra boundary points, no other point would be considered a boundary points. So it's okay for us to look at the boundary in a more intuitive fashion. We're not going to prove any of these fact. I want you to get an intuitive understanding of what it means to be open, closed, and with the boundaries. This one is unsettling. The plane itself is also closed, so it doesn't really have any boundary points, and because it doesn't have any boundary points, you could say it contains them all. Again, that's a mathematical technicality. The empty set is also closed because it contains all of its boundary points. It doesn't have any points, so it doesn't have any boundary points, so it contains them all. Again, we're being a little nitpicky but technically, the whole plane and the empty set is closed. So those are sets that are both open and closed. That is unsettling because open and closed in mathematics are not quite the same as open and close for a door or for a drawer. A door can be opened or it can be close, but it can't really be both open and closed at the same time. Sets in the complex plane or sets otherwise can be both open and closed, and there are even sets that are neither open nor closed. Open and closed is not the same meaning as it has in our daily life language. Here's a set that is neither open nor closed. The set is disc of radius r centered at z0, but now I'm including parts of the boundary, but only those parts for which the imaginary part of z minus z0 is greater than 0. So I'm looking at this disk of radius r, centered at a point z0, that's is first set. Then I'm including those points on the boundary for which the imaginary part is greater than zero. So these points are also part of my set, but these points down here are not. So since not all boundary points are included, the set is not closed, but it's also not open because for points up here, I cannot find a disk around such a point that is entirely contained in my set because part of it will always be sticking out. So the terminology of open and closed is a little bit tricky, but that's all right. We just want a vague understanding of what open and closed means, you don't have to have a perfect understanding for this course. We can force this set to be closed by closing it, and to do so, you take the closure of this set. The closure of this set, is the set and you just throw in all of its boundary points. So if you're missing some boundary points, you're adding them in right here. So the closure of a set denoted by a bar on top of this set, a kind of similar to our complex conjugate bar. So the closure of a set is a set together with all its boundary points, and then by definition has to be closed. On the other hand, we can force a set to be opened by just taking away stuff from it. The interior of a set denoted with a little circle on top of the set, is the set of all interior points. So we're taking away points that were not interior points. An example is again, the closure of the disk of radius R centered at zero, is the disk of radius R centered at zero together with a circle, so we're adding in all the boundary points. The closure of the circle is the circle itself because it already contains all of its boundary, no extra boundary points could be added to it, and so there's nothing that we have to do. If I were to take the disc of radius R and take its center away, and then take the closure, the closure would bring that centered back in, because it would happen a boundary point previously. So that's center comes back in as does the boundary. If I look at the set of all z, for which z minus z_0 is less than or equal to r. So it's kind of the disc together with a circle. If I want to take the interior of that, I would throw away the boundary points, and then get the disc of radius R, back. On the other hand, if I start only with a circle, the circle of radius R does not have any interior points. There is no point on the circle of radius R that I could find around which I can find a disk that actually fits into the circle. There's always stuff outside of the circle in there, and so it has no interior points. So the interior of that set is actually an empty set. We also need the notion of connectedness. Intuitively, a set is connected if it's in one piece. But if we want to do Mathematics with this, we need to have a more precise definition. So how do we make this more precise? What turns out being connected is difficult to make precise? So we first make it precise what it means to be separated. We say that two sets in the complex plane are separated, if there are disjoint open sets U and V, such that one of those sets, so X is contained in U and the other one is contained in V. Disjoint means that these sets have nothing in common. So U intersect V is equal to the empty set. That's what it means for these two sets to be disjoint. Before we even continue with this definition, let me give you an example. Suppose we're looking at the set X, which is the integral from 0, and 0 is included to one excluded, and Y is the set from one to two, one excluded and two included. Then these sets I claim are separated because we can find open set the U and V that are disjoint and contain U and V. So for example, you could be the disk of radius one centered at the origin. This set U contains all of X because that boundary point here was not part of X, I'm not missing anything. So it's an open set that contains all of X. On the other hand, the disk of radius one centered at two contains all of Y, and they don't intersect, they seem to touch their up because they're open disk, they actually have nothing, no point in common. So I found two open sets that contain x and y. They have nothing to do with each other. That makes X and Y separated. Now we say that a set is connected if it's impossible to find two separated non-empty sets whose union equal to W. So if I cannot find such a way of separating my set, then it's connected. So in particular, in this example, X union Y, which is really the integral from 0-2 but we're missing the point one is not connected because I can separate it. So it seems to be easier to check whether a set is not connected, but it is hard to check whether a set is connected. For open sets, thankfully, there's a much easier criterion to check whether or not the set is connected. This is only true and I can't explain them. So here's a theorem and we're not going to prove this theorem, we'll just use it. Suppose G is an open set in the complex plane, this set is connected if and only if any two points in G can be joined by successive line segments. So if I draw a set here, and the set could have some holes, so maybe it has a hole right here and it has a hole here, and maybe missing some stuff in there. But it's an open set and it's connected that can join any two points by a successive line segment. So of course, I'm picking two random points, this one and that one. So can I join them with line segments? Well, yes I can. I can just start drawing and go turn whenever I need to. I made it. So I joined them in G, so I never left my set G, and I joined them with line segments that never left G. Therefore, this theorem tells me if I can do this for any two points in G that my set G is connected. On the other hand if G consists of several parts and I picked one point over here and another one over here. I now tried to connect them with lines segments. I would have to get across and that's leaving my set. So this set, G here, is not connected, whereas this one is connected. Finally, we say a set is bounded. Why but finishes stops? So if there exists a number R such that the set is contained in a large enough ball of radius R. So let's draw the complex plane here and some set and it could be complicated set, but it stops going out toward infinity again. So if I choose a large enough radius, and that has to be pretty darn large radius, but I choose a large enough radius, this is a circle of radius R centered at the origin. Eventually, my radius is large enough so it'll contain that set. On the other hand is set for which I cannot find such an R, is called unbounded. So what are examples of unbounded sets? Here's one example. Suppose I take two radial segments and A is the set between these two radial segments. Well, that's keeps going on and go on and go on and go on no matter what radius I choose, I'm just not going to capture it all because it just keeps going. So this set would be an example of an unbounded set. Another example of an unbounded set is a whole half line that say, all those points for which the real part is positive, that's an unbounded set. The whole complex plane itself is an unbounded set. So there are plenty of unbounded sets. We already brought up this notion of infinity a little bit for those unbounded sets for which that reaching out towards infinity. In R, we remember there are two directions that give rise to infinity. I could go in the positive direction and go towards plus infinity, or it could go in the negative direction and go towards negative infinity. So 1, 2, 3, 4, 5 walkout towards plus infinity and -1,-2, -3, -4, -5, they go towards negative infinity. So we have a plus infinity and we have a negative infinity. In the complex plane, there is only one infinity. We regard infinity as no matter how you get there. If you leave any bounded set, you're going to get toward infinity. So there's lots and lots of ways of getting there. So 1, 2, 3 is one way of getting towards infinity, or -1, -2, -3, -4, -5 also goes towards infinity, but you can also go on the imaginary axis or on the negative imaginary axis, or you could even go in circles. So 1, 2i, -3, -4i, 5, 6i. So this sequence would also one off towards infinity because it leaves any bounded set. So as soon as you leave any bounded set, we'll go in toward infinity. So infinity is all out here outside of large discs we're getting closer and closer to infinity, no matter in which direction I go.