Chapter 1: Section 12: Sets in the complex plane

Question 1

$\epsilon$-neighbourhood of a complex number $z_0$ defined as a set of the form

Answer 1

$z \in \mathbb{C} : |z-z_0| < \epsilon $

for $\epsilon \in \mathbb{R}, \epsilon > 0 $

Question 2

An $\epsilon$-neighbourhood of a complex number $z_0$

$z \in \mathbb{C} : |z-z_0| < \epsilon $

is also referred to as

Answer 2

the open disc with centre $z_0$ and radius $\epsilon$

we write $D(z_0,\epsilon) = {z \in \mathbb{C} : |z-z_0| < \epsilon} $

Question 3

A deleted neighbourhood of $z_0$ is a set of the form

Answer 3

$D’(z_o,\epsilon) = {z \in \mathbf{C} : 0 < |z-z_0| < \epsilon}$

NB notice $0 < |z-z_0|$ which differs from the reqular epsilion neighbourhood, here the distance cannot be zero , i.e. the centre is deleted

Question 4

Closed disk

Answer 4

$\bar D (z_0,\epsilon)$

Includes centre and boundary points

Question 5

Assuming S is set $ \in \mathbb{C} $ and $z_0 \in \mathbb{C}$

then what is an interior point

Answer 5

$z_0$ is an interior point of S if $z_0 \in S$ and

there exsists a neghbourhood of $z_0$ that is contained in S

Question 6

Assuming S is set $ \in \mathbb{C} $ and $z_0 \in \mathbb{C}$

then what is a boundary point

Answer 6

$z_0$ is a boundary point of S if every neighbourhood of $z_0$ conatins a point of S as well as a point NOT in S

The boundary of S consists of all the boundary points of S

Question 7

Set S is open

Answer 7

S does not contain any of its boundary points

Question 8

Set S is closed

Answer 8

S contains all its boundary points

Question 9

Closure of set S

Answer 9

The closed set consisting of S together with all its boundary points

Question 10

Conenct set S

Answer 10

if any two poitns $a,b \in S$ can be joined wiht a polygonal line/ finite # of line segments that lie in S

Question 11

When is a set a domain

Answer 11

When it is

1. non-empty
2. open
3. connected

Question 12

when is a set bounded

Answer 12

when there exists a postive number R st S is contained in $D(0,R)$

Recall that this is the open disk with center 0 and radius R

Question 13

When is point $z_0$ an accumulation point/cluster point/limit point of aset S

Answer 13

if every deleted neighbourhood of $z_0$ contains a point of S

Question 14

Theorem telling us when S is open/closed

Answer 14

S is open iff
1. each of its points is an interior pt

S is closed iff
1. S contained all its accumulation points