0
votes
0answers
42 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...
1
vote
1answer
51 views

Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on ...
0
votes
1answer
29 views

Neighbourhood of a disc

I'm a bit confused on how to write down precisely a neighborhood on an example. My question is the following: Suppose I have a disc $\Omega=\lbrace x\in\mathbb{C}, |x-1|<2.5\rbrace$ and its ...
2
votes
1answer
50 views

Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
2
votes
2answers
50 views

About Path-connected

Let $a=(a_1,a_2,...,a_k)$ and $b=(b_1,b_2,...,b_k)$ be points in k-dimensional space $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function on the unit interval $[0,1]$ with values in ...
0
votes
0answers
45 views

Complex analysis winding number

We have that $f:\mathbb S^1 \rightarrow \mathbb S^1 $ and $f(z)=f(1)\widehat{\phi}(z)$ with $\widehat{\phi}(\exp{2\pi it}) = \exp(2 \pi i \phi(t)),$ where $\phi:I \rightarrow \mathbb{R} $ is a ...
1
vote
0answers
12 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
0
votes
2answers
34 views

Understanding a Similarity Transformation with a Fixed Point from the Definition of the Derivative for Complex Functions

My book has given me the definition of the derivative for complex functions in several ways but points to one in particular for its geometric aid, I quote: In this formulation ...
0
votes
1answer
21 views

Proving that the $C_b(M)$ is a complete space with the $L^{\infty}$ norm.

Suppose $A$ is some metric space, and let us define $C_b(M)$ as the vector space consisting of the set of all bounded continuous $\mathbb{R}$ valued functions on $A$. Now, we define the $L^{\infty}$ ...
0
votes
2answers
41 views

Showing that the closure of a totally bounded set is totally bounded

I would like to show that if I have a metric space $(M,d)$ and that $M$ is totally bounded, that its closure $cl(M)$ is also bounded. My general strategy is to show that for every $\epsilon > 0$, ...
0
votes
1answer
26 views

Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function

In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that ...
0
votes
0answers
18 views

Uniform convergence of $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk.

I have the sequence $f_n(z)=\sum_{j=o}^n z^j$ on the open unit complex disk ($\Delta$). My question is whether or not my approach is correct to the following problems: is the sequence normal? Does ...
1
vote
0answers
68 views

Hanging a picture with Beta functions

There's a classic puzzle that goes something like this: You have two nails in a wall, and you want to hang a picture with a string (think of a necklace with a pendant) in such a way that if you ...
0
votes
1answer
75 views

Does maximal principle imply open mapping theorem for any continuous function?

At first I spent a lot of time looking for counterexamples because I had never seen such a claim that M.P. implies O.M.T.. But later I realized the claim might be true, so I just had a try and proved ...
0
votes
1answer
61 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
1
vote
1answer
66 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
3
votes
3answers
72 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
1
vote
0answers
20 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
1
vote
0answers
37 views

Does anyone know an example of a local homeomorphism from the open unit disc onto itself that is not a homeomorphism? [duplicate]

I am interested in an example of a local homeomorphism from the open unit disc D onto itself which is not a homeomorphism. Or, could one prove that any such map is a homeomorphism?
2
votes
0answers
49 views

Why does the space of germs construction correspond to the gluing construction of Riemann surfaces?

I know this might be too broad / vague a question, but still looking for somebody to write something meaningful about this. When constructing the Riemann surfaces, why does the space of germs ...
2
votes
1answer
97 views

What are the things I need to know to study Topology

I am looking to learn about Riemann Surfaces but I know that beforehand I need to study certain subjects like Metric and Topological Spaces, Complex/Real Analysis and Complex Functions. Can anyone ...
1
vote
0answers
34 views

What “topological setting” would make complex analysis fluent?

In measure theory, the order topology on $\overline{\mathbb{R}}$ (extended real) and $[0,\infty]$ provides rich foundation to analyze measurable functions and abstract integral. Just like this, i ...
0
votes
0answers
34 views

How do i show that the Riemann sphere and the one-point compactification of $\mathbb{C}$ are homeomorphic?

Let $\mathbb{C}\cup\{\infty\}$ be the one-point compactification of $\mathbb{C}$. Let $S^2$ denote the 2-sphere in $\mathbb{R}^3$. Defne $\zeta:S^2\rightarrow \mathbb{C}\cup\{\infty\}$ as ...
2
votes
1answer
44 views

Let C be a circle. Show that the only subset of C homeomorphic to a circle is C itself

I am trying to answer the question stated in the title. The hint in my book says to realize that for any z on the circle C{z} is still connected. I believe I can deal with case that shows that C{z} ...
3
votes
1answer
54 views

Show $\Omega$ is simply connected if every harmonic function has a conjugate

Prove: If every harmonic function on $\Omega$ has a harmonic conjugate on $\Omega$, then $\Omega$ is simply connected. Same question is asked here but no proof is presented: is the converse true: in ...
1
vote
1answer
45 views

Complex Analysis analytic function

If $f$ is an analytic function on a domain $D$ and $\mathrm{Im} f$ takes on only the value $71$ then for some constant $C \in \Bbb{R}$, is it true that $f = C + 71 i$ on $D$?
0
votes
3answers
118 views

If $f$ is holomorphic and $\lvert f\rvert$ is constant, then $f$ is constant.

Let $\Omega$ be a connected open set and $f:\Omega\rightarrow\mathbb{C}$ is holomorphic. If $\lvert f\rvert$ is a constant function then we need to show, $f$ is also a constant function. I tried to ...
1
vote
1answer
142 views

Complex plane Riemann Sphere topology

Came across the following statement: Define $B_\infty(a;r)$ be the ball in $C_\infty$ with respect to the metric $d_\infty(z_1,z_2) = \frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2}\sqrt{1+|z_2|^2}}$, show that ...
2
votes
1answer
52 views

Compact Function Set

If a uniformly equicontinuous family of functions is analytic on an open disk in the complex plane, it has compact closure by Montel's theorem (and Arzela-Ascoli). Is it possible that this set is ...
1
vote
1answer
45 views

Complex plane Riemann sphere equivalent metrics?

Define $d(z_1, z_2) = |z_1 - z_2|$ and $d_\infty(z_1,z_2) = \frac{2|z_1 - z_2|}{\sqrt{1+|z_1|^2}\sqrt{1+|z_2|^2}}$. How to show the following statement? For every $a \in C$ and $r > 0$, there ...
1
vote
1answer
70 views

Why only one unbounded connected component

Here on page 344 it is stated that If $U \subset \mathbb C$ is bounded then $\mathbb C \setminus U$ has exactly one unbounded component. While it seems sort of clear to me in an intuitive way I ...
1
vote
2answers
100 views

Regions determined by a closed curve

Question: is "Idea" below flawed? Let $\gamma$ be a closed curve in the complex plane. It may intersect itself, and required only to be continuous (no differentiability assumptions). The image of ...
2
votes
3answers
50 views

The closure of a complex Set of Powers.

I came across this question while doing some exercises in complex analysis: For fixed $x\in[0,1)\setminus \mathbb{Q}$ let $a=e^{2\pi ix} \in\mathbb{C}_{|z|=1}$ and define: ...
1
vote
1answer
94 views

Connectedness at a simple boundary point

Interested by this question in math.SE, which shares a link to planetmath about definition of a simple boundary point. This link gives reference to the book Functions of one complex variable II of ...
6
votes
4answers
178 views

Simply Connected domains.

If $U$ and $U'$ be two domains in $\Bbb C$, and $f$ be a homeomorphism in $U$ and $U'$ then domain $U$ is simply connected $\iff$ $U'$ is simply connected. I found this problem in complex analysis. So ...
0
votes
1answer
43 views

Generated Algebra Closed under Conjugation

Let $\mathscr B$ be a set of continuous, complex-valued functions over some topological space $X$. Let $\mathscr G$ be the algebra generated by $\mathscr B$, i.e., the smallest algebra containing ...
4
votes
1answer
84 views

The sheaf $\mathfrak{S}$ of germs of analytic functions over $D$ is a topological group (Ahlfors)

In Ahlfors' complex analysis text, page 286 he gives the following definition: Definition 1. A sheaf over $D$ is a topological space $\mathfrak S$ and a mapping $\pi:\mathfrak S \to D$ with the ...
0
votes
1answer
26 views

Exponential of a complex line

Is there an "elementary" way to prove that if $D$ is a one-dimensional vector space in $\mathbb{C}$ (considered here as a real vector space), then $\exp(D) \neq \mathbb{C}^{\ast}$ ?
2
votes
3answers
95 views

Is $xy = 1$ connected ? [duplicate]

The graph of $xy = 1$ is connected in $\mathbb{C}^2$. The above statement is true. Why? Please show reason. In $\mathbb{R}^2$ $xy = 1$ is not connected as it has two disjoint components in $1$-st ...
2
votes
1answer
169 views

Continuous bijection between open simply connected subsets of $\mathbb{C}$

Suppose $U,V \subseteq \mathbb{C}$ are open sets. I did a proof saying if $U$ and $V$ were conformally equivalent then $U$ simply connected implies $V$ is as well. I did this by showing the ...
1
vote
1answer
62 views

Conformal Maps and Homeomorphisms

Is every conformal map from an open subset $U\subseteq\mathbb{C}$ to an open subset $V$ a homeomorphism? Here is why I think it is. A conformal map is holomorphic (hence continuous and open) and ...
5
votes
1answer
96 views

Is image of boundary a boundary of image?

I have a question which appers in problems concerning Möbius transformation, for example Let $A=\{ z\in \mathbb{C} : \|z\| <1, \Re(z)>0\}$ and $f(z)=\frac{z+i}{z-1}$ Determine $f(A)$. Often ...
4
votes
2answers
118 views

Topology of the completed upper-half plane

Define the topology on $\mathbb{H}^* : = \mathbb{H} ∪ \mathbb{Q} ∪\infty$ by taking a basis of open sets around $\infty$ to be $S_{\epsilon} : = \{ z ∈ H : Im ( z ) > 1 /\epsilon \}∪\infty$ , and ...
2
votes
1answer
69 views

The existence of “arbitrary large” connected compact sets in the plane

Studying some complex analysis I came up with the following hypothesis: Let $\Omega \subseteq \mathbb C$ be a region (an open and connected set), and let $E \subset \Omega$ be a compact subset. ...
1
vote
2answers
139 views

What does $e^z,|z|=1$ look like?

The origin is to find the connected component of $e^z,|z|=1$, as $|z|=1$ is connected,and $e^z$ is continuous, $e^z$ should be a connect and the number is $1.$ I'm concerned with the image of ...
0
votes
1answer
57 views

Differentiable functions on closed and open sets in $\mathbb{C}$

Is there a difference between functions holomorphic (on open sets $\Omega$) and functions that have derivatives everywhere on $\mathcal{Cl}(\Omega)$ (their closure in $\mathbb{C} \cup \{\infty\}$, ...
1
vote
2answers
77 views

Proof of connection between two complex set

This is an example in my undergraduate complex analysis textbook Let $S$ be the open set consisting of all points $z$ such that $|z|<1$ or $|z-2|<1$. State why $S$ is not connected. ...
1
vote
3answers
102 views

Finite number of points inside a disk

Let $n\ge 2$ and suppose that $z_1, z_2, \ldots, z_n$ are distinct points in the interior of some disk $D$ in the plane. Why is it true that there exists a smaller disk $D'\subseteq D$ such ...
2
votes
1answer
76 views

$\mathbb C\cup\{\infty\}$ is compact, a “direct proof”.

Consider the Riemann sphere $\mathbb C\cup\{\infty\}$ equipped with the usual topology. In most textbooks the compactness of $\mathbb C\cup\{\infty\}$ is proven by showing an explicit homeomorphism ...
-1
votes
1answer
84 views

Does Rouché's theorem work in real analysis?

I think if Rouché's theorem (Wikipedia) works in real analysis, then maybe we can give a simple proof of the invariant domain theorem (Wikipedia).