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,