05 - Topology in the complex plane
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welcome back and welcome to lecture five
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in our course analysis of a complex kind
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today we'll talk a little bit about
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topology in the plane topology is the
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study of shapes and we need to name some
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shapes and find out how to write them
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down in complex notation and learn about
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some concepts of topology in order to
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move to some really neat applications
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next week so let's start with something
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very simple circles in the complex plane
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a circle needs a center and a radius so
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let's call the center Z zero equals x
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zero plus iy is zero so suppose we
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wanted to write down a circle of centre
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Z zero and of radius R the inside of
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that circle we'll call that a disk so
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this inside here is the disk of radius R
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centered at Z zero and the circle itself
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will call that the circle set the
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boundaries so by B R of Z 0 V to note
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the inside the disk of radius R centered
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at Z 0 and kr is the actual circle so Z
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has distance less than R from Z 0 or Z
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has distance R from Z 0 that's kind of
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long to write is there any better way to
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write down what the distance is between
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Z and Z 0 so how do we measure distance
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in the complex plane let's draw a
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picture suppose I have a point Z 0 with
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coordinates X 0 plus iy 0 and another
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point Z with coordinates X plus iy and I
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want to find the distance between these
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two points in particular I want to
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determine whether this distance is at
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most R or equal to R so what do I know I
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know this is X 0 and this is the x
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coordinate X and similarly I know this
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is y 0 and this is
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I what is the distance between Z and Z
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zero it's the length of the connecting
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line segment between Z 0 and Z how do I
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find that while I can draw a right
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triangle here and I find a triangle with
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this right angle and so in order to find
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this distance I call that D I could just
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as well find the length of the two legs
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and use the Pythagorean theorem so how
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long are these two legs
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well this leg down here has length X
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minus X 0 because that's the distance
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between the two x-coordinates and this
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leg here has length y minus y 0 so in
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order to find the distance D I can use
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the Pythagorean theorem and find that D
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is equal to the square root of the sum
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of the squares of the two legs so the
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square of X minus x0 and the square of Y
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minus y0 but that looks awfully like the
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modulus of a complex number namely the
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complex number X minus x0 plus I times y
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minus y0 that's how we would calculate
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its modulus and this complex number is
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really the complex number Z minus YZ
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here so this distance D is actually the
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modulus of the complex number z minus z0
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and with that new notation I can write
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down the disk of radius R centered at Z
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0 in a much easier way it's the set of
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all those disease for which the modulus
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of Z minus Z 0 is less than R and the
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circle of radius R centered at Z 0 is
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the set of all C's for which the modulus
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of Z minus e 0 is equal to R meet some
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more definitions suppose were given a
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set e so some set maybe it looks like
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this maybe it has a hole who knows we
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say an interior point of E is a point
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like this Point C is 0 such that there
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exists some R such that the disc of
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radius R
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zero is entirely contained in E so I can
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draw a little disk around this point and
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it still fits entirely into the set E so
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obviously if I try to do that out here
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so if I try at this point like no matter
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how small a disk I draw it's never going
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to be contained in E but even will point
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like this is if I draw a disk around
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that some part of that this will always
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stick out or a point right here on that
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edge if it's right on the edge no matter
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how small I draw the disk it will always
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stick out so interior points are points
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that are really inside of E and there's
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some rooms on buffer room before I
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really had the end of e so those are
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interior points of weak points like this
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are called
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boundary points of e so a point B is on
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the boundary of e in every disk so that
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draw around it contains some points in E
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so let's write good one so I didn't need
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some points in E but also some points
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that are not in E so no matter our house
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wall I choose this disk I will always
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capture some points in E and some
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outside of E those points are called
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boundary points of E and the boundary is
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denoted with this funny new Greek symbol
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that we had not introduced before it's
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like a Delta symbol almost so that's a
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symbol for the boundary of E
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we say that a set is open if every one
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of its points is an interior point and
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on the other hand we say is that is
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closed if it contains all of its
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boundary points so those are actually
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quite subtle definitions so we need to
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look at some examples to understand
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those a little bit better so the ball of
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radius R centered at C 0 is an example
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of an open set let's look at that a disc
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Center data points is 0 and our radius R
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I'm climbing that's an open set so no
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matter which point you pick I can always
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find a little disc around that point it
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still fits into the original big disc
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even if it's really close to the H I
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can't be on the edge because points on
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the edge are not included in my disk of
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radius R because I require the distance
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between Z and Z or to be strictly less
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than R so no matter where I am I'm not
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quite on the boundary and so there's
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always room for a little disc around it
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without hitting the boundary also
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everything out here is also all those
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points for which Z minus e 0 is bigger
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than R that's also open because I can
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find for any point a little disc around
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it that stays entirely outside of the
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disc of radius R the whole plane is open
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well you know what every point is
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interior point in the whole plane
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because no matter where I put a point in
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the plane you can find the disc around
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that it fits into the complex plane so
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that's kind of trivial the empty set
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being open is a little funky but the
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empty sets doesn't have any interior
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points and so it contains all of those
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so that's a little bit of a mathematical
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technicality that the empty set is also
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open now here's an example of a closed
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set the disc of radius R but this time
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instead of writing strict inequality I
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made it a less than or equal to so I did
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include these boundary point
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now that time no it's a closed set
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because it's the disc together with all
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of its boundary points so this includes
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all of its boundary points and it's a
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closed set the circle itself is also
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closed there's no extra boundary points
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no other point would be considered a
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boundary point so it's okay for us to
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look at the boundary in a more intuitive
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fashion we're not going to prove any of
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these fact I want you to get an
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intuitive understanding of what it means
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to be open closed and with the
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boundaries this one is kind of
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unsettling the plane itself is also
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closed it doesn't really have any
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boundary points because it doesn't have
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any boundary points you could say it
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contains them all again that's a
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mathematical technicality the empty set
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is also closed because it contains all
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of its boundary points it doesn't have
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any points so it doesn't have any
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founder going so it contains them all
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again we're being a little nitpicky but
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technically the whole plane and the
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empty set are closed so those are sets
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that are both open and closed that is
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unsettling because open and closed in
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mathematics are not quite the same as
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open and closed for a door or for a
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drawer a door can be open or it can be
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closed but it can't really be both open
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and closed at the same time sex in the
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complex plane or sets otherwise can be
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both open and closed and there are even
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sets that are neither open nor close
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open and close is not the same meaning
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as it has in our daily life language
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here's a set that is neither open nor
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closed the set is the disc of radius R
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centered at to zero but now I'm
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including parts of the boundary but only
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those parts for which the imaginary part
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of Z minus e0 is greater than zero so
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I'm looking at this disc of radius R
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centered at a point Z 0 that's this
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first set and then I'm including those
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points on the boundary for which the
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imaginary part is greater than 0 so
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these points are also part of my set but
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these
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points down here are not and so since
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not all boundary points are included
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this set is not closed but is also not
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open because four points up here I
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cannot find a disk around such a point
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that is entirely contained in my set
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because part of it will always be
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sticking out from the terminology of
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open and closed is a little bit tricky
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but that's alright
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we just want a vague understanding of
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what open and closed means you don't
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have to have a perfect understanding for
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this course
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we can force it set to be closed by
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closing it and to do so you take the
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closure of the set the closure of a set
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is the set and you just throw in all of
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its boundary points so if you're missing
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some boundary points you're adding them
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in right here so the closure of a set
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denoted by a bar on top of this that
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kind of similar to our complex conjugate
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bar so if the closure of a set is that
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set together with all its boundary
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points and then my definition has to be
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closed on the other end again force is
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set to be open by just taking a waist F
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from it the interior of a set denoted
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with a little circle on top of the set
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is the set of all interior points so
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we're taking away points that were not
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interior points and examples again the
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closure of the disc of radius R centered
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at C 0 is the disc of radius RS and it
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is e zero together with a circle so
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we're adding in all the boundary points
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the closure of the circle is the circle
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itself because it already contains all
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of its battery no extra boundary points
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could be added to it and so there's
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nothing that we have to do if I were to
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take the disk of radius R and take its
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center away and then take the closure
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the closure would bring that Center back
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in because it would happen a boundary
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point previously so that Center comes
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back in ask that's the boundary if I
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look at the set of all Z for which Z
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minus e0 is less than or equal to R so
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it's kind of the disk together with a
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circle if I want to take the interior of
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that I would throw away the boundary
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points and then get the disk of radius R
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back on the other hand if I start only
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with a circle the circle of radius R
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does not have any interior points
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there's no point on the circle of radius
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R that I could find around which I can
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find a disc that actually fits into the
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circle is always stopped outside of the
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circle in there and so it has no
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interior points so the interior of that
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set is actually the empty set we also
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need the notion
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connected intuitively isettas connected
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if it's kind of in one piece but if we
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want to do mathematics with this we need
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to have a more precise definition so how
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do we make this more precise what turns
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out being connected is difficult to make
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precise so we first make it precise what
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it means to be separated we say that two
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sets in the complex plane are separated
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if there are disjoint open sets U and V
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such that one of those sets CX is
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contained in U and the other one is
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contained in V this joint means that
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these sets have nothing in common so you
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intersect V is equal to the empty set
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that's what it means for these two sets
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to be disjoint before we could even
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continue with this definition let me
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give you an example suppose we're
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looking at the set X which is the
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integral from zero and zero is included
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to one with one excluded and Y is the
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set from one to two one excluded and to
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include it
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then these sets I claim are separated
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because we can find open set the U and V
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that are disjoint and contain hewing V
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so for example you could be the disc of
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radius one centered at the origin this
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set u contains all of X because that
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boundary point here was not part of X
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I'm not missing anything that's an open
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set that contains all of X and on the
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other hand the disc of radius one
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centered at two contains all of Y and
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they don't intersect they kind of seem
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to touch there up because they're open
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discs they actually have nothing no
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point in common so I found two open sets
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that contain X and y they have nothing
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to do with each other and that makes the
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x and y
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right now we say that a set is connected
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if it's impossible to find two separated
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non-empty sets whose union equal to W so
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if I cannot find such a way of
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separating my set then it's connected so
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in particular in this example X Union Y
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which is really the interval from 0 to 2
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but we're missing the point 1 is not
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connected because I can separate it so
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it seems to be easier to check whether a
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set is not connected but it is hard to
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check whether a set is connected for
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open sets thankfully there's a much
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easier criterion to check whether or not
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the set is connected this is only true
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in the complex plane though so here's a
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theorem we're not going to prove this
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there and we'll just use it suppose G is
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an open set in the complex plane
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this set is connected if and only if any
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two points in G can be joined by
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successive line segments so if I draw a
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set here and the set could have some
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holes so maybe it has a hole right here
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and as a hole here and maybe missing
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some stuff there but it's an open set
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and it's connected they can join any two
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points by a successive line segments of
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course I'm picking two random points
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this one and that one so can I join them
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with line segments or yes I can you can
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just start drawing and go turn whenever
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I need to and I made it so I joined them
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in G so I never left my set G and I
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joined the would line segments that
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never left G and therefore this theorem
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tells me if I can do this for any two
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points in G that my set G is connected
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on the other hand if G consists of sort
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of several parts and I picked one point
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over here and another one over here and
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I now try to connect them with line
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segments I would kind of have to get it
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cross and that's leaving my set
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so this set G here is not connected
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whereas this one is connected
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finally we see a said us-bound it why
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the kind of finishes stops so if there
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exists a number R such that the set is
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contained in a large enough ball of
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radius R so let's draw a complex plane
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here and some set and it could be a
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complicated set but it stops going out
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toward infinity kind of so if I choose
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to large enough radius and it has to be
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pretty darn large radius but let's use a
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large enough radius this is a circle of
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radius R centered at the origin
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eventually my ratings with large enough
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so it will contain that set on the other
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hand is set for which I cannot find such
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an R
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it's called unbounded so what are
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examples of unbounded sets here's one
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example suppose I take two radial
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segments and a is the set between these
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two radial segments well that keeps
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going and going and go on and go on no
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matter what radius I choose I'm just not
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going to capture at all because it just
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keeps going so this set would be an
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example of an unbounded set another
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example of an unbounded set is a whole
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half line let's say all those points for
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which the real part is positive that's
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an unbounded set the whole complex plane
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itself is an unbounded set so there are
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plenty of unbounded sets we already
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brought up this notion of infinity a
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little bit for those unbounded sets for
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which they're kind of reaching out
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towards infinity in R we remember there
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are two directions that give rise to
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infinity I could go in the positive
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direction and go towards plus infinity
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or I could go in the negative direction
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and go towards negative infinity so you
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know 1 2 3 4 or 5 walk out towards plus
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infinity and minus 1 - 2 - 3 - 4 - 5
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they go toward negative infinity so we
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have a plus infinity and we have a
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negative infinity in the complex plane
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that only one infinity we regard
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infinity as no matter how you get there
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if you
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leave any bound it's said you're going
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to get toward infinity so there's lots
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and lots of ways of getting there so one
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two three is one way of getting towards
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infinity or - one - two - three - four -
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five that also goes toward infinite but
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you can also go on the imaginary axis or
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on the negative imaginary axis or you
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could even go in circles so 1 2 I minus
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3 minus 4 I 5 6
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I so this sequence would also run off
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towards infinity because it leaves any
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bounded set so as soon as you leave any
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bounded set we're going toward infinity
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so infinity is sort of all out here
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outside of large disks we're getting
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close on closer to infinity no matter in
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which direction I go