September18 ,2020 TOPOLOGYOFTHECOMPLEXPLANE ComplexVariables

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MAT334H1-F – LEC0101

Complex Variables

TOPOLOGY OF THE COMPLEX PLANE

September 18^th, 2020

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Today’s topic: topology ofℂ

After having tried to convince you that a line is a circle, today I will divide by 0…

LEC0101 website: http://uoft.me/MAT334-LEC0101

How to practice for MAT334:

• Immediate practice questions from the slides.

• ”Problems_to_…” from Quercus.

• Tutorial problems.

Make sure that you have read the Outline/Syllabus andreadmepages posted on Quercus, they contain valuable information such as:

Each Quiz is drawn from the problems for a week (or weeks) in the Quiz description.

(Problems_to_…).

Bewareif you look online forinversion: it can mean the complex inversion𝑧 ↦ 𝑧⁻¹(we use this one) but also the geometric inversion, which in terms of complex numbers is given by𝑧 ↦ 𝑧⁻¹, do NOT confuse them!

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Disks, neighborhoods and bounded sets

Open disk

Theopen disk centered at𝑧₀∈ ℂand of radius𝑟 ∈ ℝ_>0is𝐷_𝑟(𝑧₀) ≔ {𝑧 ∈ ℂ ∶ |𝑧 − 𝑧0| < 𝑟}. (You may also see the name𝑟-vicinity of𝑧₀.)

Closed disk

Theclosed disk centered at𝑧₀∈ ℂand of radius𝑟 ∈ ℝ_>0is𝐷_𝑟(𝑧₀) ≔ {𝑧 ∈ ℂ ∶ |𝑧 − 𝑧0| ≤ 𝑟}.

Neighborhood

We say that𝑆 ⊂ ℂis aneighborhoodof𝑧₀∈ ℂif there exists𝜀 ∈ ℝ_>0 such that𝐷_𝜀(𝑧₀) ⊂ 𝑆.

Bounded sets

We say that𝑆 ⊂ ℂisboundedif there exists𝑟 ∈ ℝ_>0such that𝑆 ⊂ 𝐷_𝑟(0).

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆

𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆 𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆 𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Definition: interior

Theinteriorof𝑆 ⊂ ℂis𝑆 ≔ {𝑧 ∈ ℂ ∶ ∃𝜀 > 0, 𝐷̊ 𝜀(𝑧) ⊂ 𝑆}(also denoted𝑆^int).

Definition: closure

Theclosureof𝑆 ⊂ ℂis𝑆 ≔ {𝑧 ∈ ℂ ∶ ∀𝜀 > 0, 𝐷𝜀(𝑧) ∩ 𝑆 ≠ ∅}.

Definition: boundary

Theboundaryof𝑆 ⊂ ℂis𝜕𝑆 ≔ 𝑆 ⧵ ̊𝑆.

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

4 𝜕𝑆 = {𝑧 ∈ ℂ ∶ ∀𝜀 > 0, 𝐷𝜀(𝑧) ∩ 𝑆 ≠ ∅and𝐷_𝜀(𝑧) ∩ 𝑆^𝑐≠ ∅}

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆

𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆 𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆 𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Definition: interior

Definition: closure

Definition: boundary

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆 𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆 𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Definition: interior

Definition: closure

Definition: boundary

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆 𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆 𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Definition: interior

Definition: closure

Definition: boundary

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆 𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆 𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆

𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Definition: interior

Definition: closure

Definition: boundary

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆 𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆 𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆

𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Definition: interior

Definition: closure

Definition: boundary

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆 𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆 𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆 𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Definition: interior

Definition: closure

Definition: boundary

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Interior, closure and boundary

𝑎

𝑐 𝑏

𝑑

𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∉ 𝑆 𝑑 ∉ 𝑆 𝑎 ∈ ̊𝑆 𝑏 ∉ ̊𝑆 𝑐 ∉ ̊𝑆 𝑑 ∉ ̊𝑆 𝑎 ∈ 𝑆 𝑏 ∈ 𝑆 𝑐 ∈ 𝑆 𝑑 ∉ 𝑆 𝑎 ∉ 𝜕𝑆 𝑏 ∈ 𝜕𝑆 𝑐 ∈ 𝜕𝑆 𝑑 ∉ 𝜕𝑆

Properties

1 𝑆 ⊂ 𝑆 ⊂ 𝑆̊

2 𝑆 = 𝑆 ∪ 𝜕𝑆

3 𝑆 ∩ 𝜕𝑆 = ∅̊

4 𝜕𝑆 = {𝑧 ∈ ℂ ∶ ∀𝜀 > 0, 𝐷𝜀(𝑧) ∩ 𝑆 ≠ ∅and𝐷_𝜀(𝑧) ∩ 𝑆^𝑐 ≠ ∅}

5 𝜕 (𝑆^𝑐) = 𝜕𝑆

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Open and closed sets

Definition: Open sets

We say that𝑆 ⊂ ℂisopenif𝑆 = 𝑆.̊

Definition: Closed sets

We say that𝑆 ⊂ ℂisclosedif𝑆 = 𝑆.

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Open and closed sets

Definition: Open sets

Definition: Closed sets

Examples

• {𝑧 ∈ ℂ ∶ |𝑧| < 1}is open not closed.

• {𝑧 ∈ ℂ ∶ |𝑧| ≤ 1}is closed not open.

• ℂis both open and closed.

• {𝑧 ∈ ℂ ∶ ℜ(𝑧) = 0, ℑ(𝑧) > 0}is neither open nor closed.

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Open and closed sets

Definition: Open sets

Definition: Closed sets

Properties

1 𝑆is open if and only if𝑆 ∩ 𝜕𝑆 = ∅

2 𝑆is open if and only if it is a neighborhood of each of its elements, i.e. ∀𝑧 ∈ 𝑆, ∃𝜀 > 0, 𝐷_𝜀(𝑧) ⊂ 𝑆

3 𝑆is closed if and only if𝜕𝑆 ⊂ 𝑆

4 𝑆is closed if and only if𝑆^𝑐is open

5 Open sets are stable by unions and finite intersections.

6 Closed sets are stable by intersections and finite unions.

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Open and closed sets

Definition: Open sets

Definition: Closed sets

Homework

Are the following sets open? closed?

1 {𝑧 ∈ ℂ ∶ ℜ(𝑧) ≥ 0}

2 {𝑧 ∈ ℂ ∶ ℜ(𝑧) > 0}

3 ∅

4 {𝑧 ∈ ℂ ∶ |𝑧| = 1}

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Connectedness – 1

Definition: Path-connectedness

A subset𝑆 ⊂ ℂispath-connectedif for any𝑧₀, 𝑧₁∈ 𝑆there exists𝛾 ∶ [0, 1] → ℂcontinuous such that ¹ ∀𝑡 ∈ [0, 1], 𝛾(𝑡) ∈ 𝑆 ² 𝛾(0) = 𝑧₀ ³ 𝛾(1) = 𝑧₁.

Theorem

Anopensubset𝑆 ⊂ ℂis path-connected if and only if for any𝑧₀, 𝑧₁∈ 𝑆there exists a polygonal curve from𝑧₀to𝑧₁which is included in𝑆,

i.e. there exists𝑤₀, … , 𝑤_𝑘such that[𝑤_𝑖, 𝑤_𝑖+1] ⊂ 𝑆,𝑤₀= 𝑧₀and𝑤_𝑘= 𝑧₁.

𝑧₀ 𝑧₁

Intuitively, it means that𝑆is made of only one piece.

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Connectedness – 1

Definition: Path-connectedness

Theorem

𝑧₀ 𝑧₁

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Connectedness – 1

Definition: Path-connectedness

Theorem

Beware

The openness assumption is very important in the previous theorem.

Indeed, the following set is path-connected but two point on it can’t be joined by a polygonal curve staying in the curve.

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Connectedness – 2

Definition: Connectedness

We say that anopensubset𝑆 ⊂ ℂisconnectedif it is path-connected.

Remark

We defined connectedness only for open sets: there exists a more general notion of connectedness but it coincides with path-connectedness for open sets.

Definition: Domain

We say that a subset𝐷 ⊂ ℂis adomainif it is open and connected.

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Convex sets and star-shaped sets

Definition: Convex sets

We say that𝑆 ⊂ ℂisconvexif∀𝑧₀, 𝑧₁∈ 𝑆, ∀𝑡 ∈ [0, 1], (1 − 𝑡)𝑧₀+ 𝑡𝑧₁∈ 𝑆.

(a)Convex set (b)Non-convex set

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Convex sets and star-shaped sets

Definition: Convex sets

Definition: Star-shaped sets

We say that𝑆 ⊂ ℂisstar-shapedif∃𝑤 ∈ 𝑆, ∀𝑧 ∈ 𝑆, ∀𝑡 ∈ [0, 1], (1 − 𝑡)𝑤 + 𝑡𝑧 ∈ 𝑆.

𝑤

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Convex sets and star-shaped sets

Definition: Convex sets

Definition: Star-shaped sets

Proposition

Convex and non-empty ⟹ star-shaped ⟹ path-connected.

Y

⟸ ⟸Y

Beware: ∅is convex but not star-shaped.

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Convex sets and star-shaped sets

Definition: Convex sets

Definition: Star-shaped sets

Proposition

Convex and non-empty ⟹ star-shaped ⟹ path-connected.

Y

⟸ ⟸Y

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 1

We set𝑆²≔ {(𝑟, 𝑠, 𝑡) ∈ ℝ³ ∶ 𝑟²+ 𝑠²+ 𝑡²= 1}

and𝑁 = (0, 0, 1)(thenorth poleof𝑆²).

We identifyℂwith the equatorial plane𝑃 = {𝑡 = 0}.

We define the stereographic projection with respect to𝑁:

𝜑 ∶{

𝑆²⧵ {𝑁} → 𝑃

𝑀 ↦ (𝑁𝑀) ∩ 𝑃

Theorem

𝜑is a bijection.

It allows to identifyℂwith𝑆²⧵ {𝑁}and then to see𝑁asthe point at infinity, i.e. to identify𝑆² withℂ = ℂ ⊔ {∞}.̂

There are several models forℂ, this one is called thê Riemann sphere.

N

M

φ(M)

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 1

We set𝑆²≔ {(𝑟, 𝑠, 𝑡) ∈ ℝ³ ∶ 𝑟²+ 𝑠²+ 𝑡²= 1}

𝜑 ∶{

𝑆²⧵ {𝑁} → 𝑃

𝑀 ↦ (𝑁𝑀) ∩ 𝑃

Theorem

𝜑is a bijection.

N

M

φ(M)

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 1

We set𝑆²≔ {(𝑟, 𝑠, 𝑡) ∈ ℝ³ ∶ 𝑟²+ 𝑠²+ 𝑡²= 1}

𝜑 ∶{

𝑆²⧵ {𝑁} → 𝑃

𝑀 ↦ (𝑁𝑀) ∩ 𝑃

Theorem

𝜑is a bijection.

N

M

φ(M)

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 2

Definition: Neighborhood of the ∞

We say that𝑉 ⊂ ℂisa neighborhood of∞if𝑉^𝑐≔ ℂ ⧵ 𝑉 is bounded.

Proposition

𝑉 ⊂ ℂis a a neighborhood of∞if and only if∃𝑅 ∈ ℝ_>0, {𝑧 ∈ ℂ ∶ |𝑧| > 𝑅} ⊂ 𝑉.

Remember that a set is open if and only if it is a neighborhood of each of its points. Hence, we defined a topology onℂ. It makeŝ 𝜑 ∶ 𝑆²→ ̂ℂa homeomorphism.

Definition: Open sets of ℂ ̂

A subset𝑆 ⊂ ̂ℂis open if

• 𝑆 ⊂ ℂis open or

• 𝑆 = {∞} ∪ 𝑈 where𝑈 = 𝐾^𝑐⊂ ℂis the complement of a compact𝐾 ⊂ ℂ(closed and bounded). The Riemann sphere is a special case of a general topological construction called theone-point

compactificationor theAlexandrov compactification.

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 2

Definition: Neighborhood of the ∞

Proposition

Definition: Open sets of ℂ ̂

• 𝑆 = {∞} ∪ 𝑈 where𝑈 = 𝐾^𝑐⊂ ℂis the complement of a compact𝐾 ⊂ ℂ(closed and bounded). The Riemann sphere is a special case of a general topological construction called theone-point

compactificationor theAlexandrov compactification.

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 2

Definition: Neighborhood of the ∞

Proposition

Definition: Open sets of ℂ ̂

• 𝑆 = {∞} ∪ 𝑈 where𝑈 = 𝐾^𝑐⊂ ℂis the complement of a compact𝐾 ⊂ ℂ(closed and bounded).

The Riemann sphere is a special case of a general topological construction called theone-point compactificationor theAlexandrov compactification.

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 2

Definition: Neighborhood of the ∞

Proposition

Definition: Open sets of ℂ ̂

• 𝑆 = {∞} ∪ 𝑈 where𝑈 = 𝐾^𝑐⊂ ℂis the complement of a compact𝐾 ⊂ ℂ(closed and bounded).

The Riemann sphere is a special case of a general topological construction called theone-point compactificationor theAlexandrov compactification.

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The extended complex plane: ℂ = ℂ ⊔ {∞} ̂ – 3

We may extend the inversion^atoℂ̂byinv∶

⎧⎪

⎨⎪

⎩

ℂ̂ → ℂ̂

𝑧 ↦ 𝑧⁻¹ if𝑧 ∈ ℂ ⧵ {0}

0 ↦ ∞

∞ ↦ 0

aActually, it is possible to define division by0, what isnotpossible is to define a multiplicative inverse of0.

Remember from last time the generalized circle (or line-circle) equation

𝑎𝑧𝑧 − 𝜂𝑧 − 𝜂𝑧 + 𝑘 = 0 (1)

where𝑎, 𝑘 ∈ ℝand𝜂 ∈ ℂsatisfy|𝜂|²− 𝑎𝑘 > 0.

Set𝑤 = 𝑧⁻¹(so that we swap0and∞inℂ).̂ Then (1) becomes

𝑘𝑤𝑤 − 𝛼𝑤 − 𝛼𝑤 + 𝑎 = 0 where𝛼 = 𝜂.

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Not part of MAT334, just a gift  :

another example of one-point/Alexandrov compactification

The drawing on the left is probably my favourite real algebraic set (Whitney umbrella) and the one on the right is its one-point/Alexandrov compactification.