# Tagged Questions

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### 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 ...
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### 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 $L^{\infty}$ ...
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### 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 $\epsilon > 0$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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} ...
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### 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 ...
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### 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 $D$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\exp(D) \neq \mathbb{C}^{\ast}$ ?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 $\mathbb{C} \cup \{\infty\}$, ...
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### 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. ...
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### 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 ...
### $\mathbb C\cup\{\infty\}$ 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 ...