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Topology of Complex Number

Complex Numbers in Complex Plane

Unlike the one-dimensional number line, the range of complex numbers in the complex plane are usually described by circles and disks. For a given complex number z₀=x₀+iy₀, take the complex number z₀ as center and consider the radius r arround, then

Open disk or neighborhood of complex numbers with radius r, and centered at z₀: Bᵣ(z₀)={z∈ℂ: z has distance less than r from z₀}
Circle of complex numbers with radius r, and centered at z₀: Kᵣ(z₀)={z∈ℂ: z has distance r from z₀} 
Closed disk of complex numbers with radius r, and centered at z₀: Dᵣ(z₀)={z∈ℂ: z has distance less than or equal r from z₀}
Radius r = distance d between two complex points = √((x-x₀)²+(y-y₀)²)=|(x-x₀)+ (y-y₀)|=|z-z₀|
⇒Bᵣ(z₀)={z∈ℂ: |z-z₀|<r} , Kᵣ(z₀)={z∈ℂ: |z-z₀|=r} , and Dᵣ(z₀)={z∈ℂ: |z-z₀|≤r}

Interior and Boundary Points

By definition, let E⊂ℂ. A point z₀ is an interior point of E if there is some r>0 such that Bᵣ(z₀)⊂E. And by definition, let E⊂ℂ. A point b is a boundary point of E if every disk around b contains a point in E and a point not in E. The boundary of the set E⊂ℂ, ϑE, is therefore the set of all boundary points of E.

Open and Closed Sets

By definition, a set U⊂ℂ is open if everyone of its points is an interior point. And by definition, a set A⊂ℂ is closed if it contains all of its boundary points.

{z∈ℂ: |z-z₀|<r} and {z∈ℂ: |z-z₀|>r} are open.
ℂ and ∅ are open
{z∈ℂ: |z-z₀|≤r} and {z∈ℂ: |z-z₀|=r} are closed.
ℂ and ∅ are closed
{z∈ℂ: |z-z₀|<r}∪{z∈ℂ: |z-z₀|=r and Im(z-z₀)>0} is neither open nor closed.

Closure and Interior of a Set

By definition, let E be a set in ℂ. the closure of E is the set E together with all of its boundary points: E̅=E∪ϑE. By definition, the interior of E, E̊ is the set of all interior points of E.

Bᵣ(z₀)=Bᵣ(z₀)∪Kᵣ(z₀)={z∈ℂ: |z-z₀|<r}
Kᵣ(z₀)=Kᵣ(z₀)
Bᵣ(z₀)\{z₀}={z∈ℂ: |z-z₀|≤r}
With E={z∈ℂ: |z-z₀|≤r}, E̊=∅
With E=Kᵣ(z₀), E̊=∅

Connectedness

Intuitively: A set is connected if it is "in one piece". By definition, two sets X, Y in ℂ are separated if there are disjoint open set U, V so that X⊂U and Y⊂V. A set W in ℂ is connected if it is impossible to find two separated non-empty sets whose union equals W,

X=[0,1) and Y=(1,2] are separated. For example, chosse U=B₁(0), V=B₁(2). Thus X∪Y=[x,2]\{1} is not connected. 
It is hard to check whether a set is connected.

For open sets, there is a much easier criterion to check whether or not a set is connected:

By Theorem. 
Let G be an open set in ℂ. Then G is connected if and only if any two points in G can be joined in G by successive line segments

Bounded Sets

By definition, a set A in ℂ is bounded if there exists a  number R>0 such that A⊂BR(0). If no such R exists then A is called unbounded.

The Point at Infinity

In ℝ, ther are two directions that give rise to ±∞. But in ℂ, there is only one ∞ which can be attained in many all directions.