Chapter 4: Definitions about sets

Question 1

DEF neighbourhood

Answer 1

A neighbourhood of z_0 ∈ C is a set of the form {z ∈ C : |z − z_0| < δ} for
some δ > 0. (i.e. it is an open disc about z_0

diagram:
Neighbourhood is st around a point open ball, circle dashed boundary around point z_0

Question 2

DEF 4.2: open set

Answer 2

A set D ⊆ C is said to be open if each point z_0 ∈ D has a neighbourhood contained in the set.
(all its boundary points are “missing”.)

Question 3

Example:
Disc
D = {z ∈ C : |z| < 1}

Answer 3

The disc D = {z ∈ C : |z| < 1} is open, since Dz0 = {z ∈ C : |z − z_0| < 1 2 (1 − |z0|)} is a neighbourhood of z_0 lying in {z ∈ C : |z| < 1} whenever |z_0| < 1.

Diagram:
Dashed unit circle with a point z_0 within the circle st neighbourhood well within circle

Question 4

DEF: connected set

conditions

Answer 4

DEF: connected set
when..
A non-empty open set D ⊆ C is connected if given any points z, w ∈ D, we can find a path γ ⊂ D with initial point z and final point w.
(i D is “all in one piece”)

Question 5

{z ∈ C : |z| < 1} ∪ {1}

Answer 5

The set {z ∈ C : |z| < 1} ∪ {1} is not open, since any neighbourhood of 1 contains points with modulus greater than 1. Such points are outside the set.

Diagram:
Dashed unit circle with a point z_0 at 1 within the SET (union) st neighbourhood IS NOT within (circle) union

Question 6

DEF: region

Answer 6

A non-empty, open, connected set is called a region.

To check whether a set is a region must check if set is :

Non-empty
open
connected

CHECK

Question 7

DEF: simply connected region

Answer 7

A region D is said to be simply connected if it has no “holes”, i.e., if every point in the interior of any simple contour in D is contained in D.

Question 8

EXAMPLE: half plane

H = {z ∈ C : Re z > a}

Answer 8

The half-plane H = {z ∈ C : Re z > a} is a non-empty, open, connected set and it is also simply connected.

Diagram:
Half plane, dashed boundary Re z =a

        |            :\\\\\\\\\\\\\\\\\\\\
        |            :\\\\\\\\\\\\\\\\\\\\
        |            :\\\\\\\\\\\\\\\\\\\\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
        |         a :\\\\\\\\\\\\\\\\\\\\
        |            :\\\\\\\\\\\\\\\\\\\\
        |            :\\\\\\\\\\\\\\\\\\\\

Question 9

EXAMPLE: annulus
A = {z : r < |z − a| < R} (with 0 < r < R)

diagram?

Answer 9

The annulus A = {z : r < |z − a| < R} (with 0 < r < R) is a non-empty, open, connected set, but it is not simply connected since the set {z ∈ C : |z − a| ≤ r} is a hole

If r is less than t is less than R the points in the interior of {z : |z − a| = t} are all points with |z − a| < t, but not all these are contained within the annulus itself.

Diagram: ring with inner radius r, outer radius R, centre a. Hole is points st |x-a| is less than r. All dashed boundaries.

Question 10

EXAMPLE: punctured plane

C{a}

Answer 10

The punctured plane C{a} is a non-empty, open, connected set but it is not simply connected.

A contour can be found with a in its interior but a isn’t in the original set.

The punctured plane is a region, but it is not a simply connected region.

Question 11

EXAMPLE:

The cut plane C∗ = C \ {z ∈ C : z is real and z ≥ 0}

Diagram

Answer 11

The cut plane C∗ = C \ {z ∈ C : z is real and z ≥ 0} is a non-empty, open, connected set and it is also simply-connected.

DIAGRAM:
All of the Complex plane, except the real axis,

y=0 points