Complex Analysis
Topology
Draft for Information Only Content Topology of Complex Number source/reference: Topology of Complex NumberComplex Numbers in Complex PlaneUnlike the onedimensional 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 = √((xx₀)²+(yy₀)²)=(xx₀)+ (yy₀)=zz₀ ⇒Bᵣ(z₀)={z∈ℂ: zz₀<r} , Kᵣ(z₀)={z∈ℂ: zz₀=r} , and Dᵣ(z₀)={z∈ℂ: zz₀≤r} Interior and Boundary PointsBy 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 SetsBy 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: zz₀<r} and {z∈ℂ: zz₀>r} are open. ℂ and ∅ are open {z∈ℂ: zz₀≤r} and {z∈ℂ: zz₀=r} are closed. ℂ and ∅ are closed {z∈ℂ: zz₀<r}∪{z∈ℂ: zz₀=r and Im(zz₀)>0} is neither open nor closed. Closure and Interior of a SetBy 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: zz₀<r} Kᵣ(z₀)=Kᵣ(z₀) Bᵣ(z₀)\{z₀}={z∈ℂ: zz₀≤r} With E={z∈ℂ: zz₀≤r}, E̊=∅ With E=Kᵣ(z₀), E̊=∅ ConnectednessIntuitively: 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 nonempty 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 SetsBy 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 InfinityIn ℝ, ther are two directions that give rise to ±∞. But in ℂ, there is only one ∞ which can be attained in many all directions.
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