0
votes
0answers
48 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
0answers
31 views

is this space dense?

we know that $C(\partial\mathbb{D})$ as the continious functions on $\partial\mathbb{D}$ is dense in $L^2(\partial\mathbb{D})$. is it true that for every n the set $\{{f(t)e^{int}: f\in ...
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
63 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
43 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 ...
0
votes
0answers
15 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
39 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
2answers
80 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
33 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
30 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 ...
0
votes
1answer
32 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
40 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
41 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
109 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
123 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
49 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
38 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
59 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
76 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
90 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
153 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
28 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 ...
3
votes
1answer
62 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
24 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
93 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
124 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
51 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
88 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
109 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
68 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
138 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
54 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
69 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
99 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
75 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
81 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).
-1
votes
1answer
79 views

The boundary of a unbounded simply connected planar domain

Let $D$ be a unbounded simply connected planar open domain. Let $\partial D$ be its boundary. The question is the following: Can $\partial D$ have a compact component?
5
votes
1answer
146 views

$f$ holomorphic from unit disc to itself. $f\left(\frac{1}{2}\right) = f\left(-\frac{1}{2}\right) = 0$. Show that $|f(0)| \le 1/3$.

I'm studying for a qual exam. I cannot solve the following problem: Let $f$ be holomorphic from the unit disc to itself. $f\left(\frac{1}{2}\right) = f\left(-\frac{1}{2}\right) = 0$. Show that ...
0
votes
2answers
116 views

How do you prove that the complex inverse is continuous?

I tried to show that the continuous at a point $\delta / \epsilon$ definition holds but failed. Now I'm thinking along the lines of multiplicative group: $C \rightarrow C, x \mapsto bx$ has inverse ...
1
vote
2answers
70 views

To argue that a point is not an accumulation point of a given set

I want to show that $\mathbb Z^2$ has no accumulation points in $\mathbb R^2\backslash\mathbb Z^2$. Is this argument correct? In particular, have I correctly invoked the density property of $\mathbb ...
0
votes
1answer
52 views

If a set in a general metric space consistes entirely of isolated points, can it still have any accumulation points in its complement?

It seems not in $\mathbb R ^n$ (correct?), but how about in a general metric space? On the other hand, I'm not so sure about my claim above regarding $\mathbb R^n$: surely you can have points outside ...
2
votes
2answers
73 views

Please help me check my metric definition of isolated point

I translated the word definitions into the more symbolic form below, but as they aren't mere negations of each other, it was a little tricky. Is there any mistake below (especially for 'isolated ...
3
votes
1answer
121 views

Is this region simply connected?

Let $ \gamma (t) = t + it^2$ be a curve in the complex plane, $0 \leq t \leq \infty$. My question is: Is $ D = \mathbb{C} \setminus \gamma $ simply connected? The curve separates the complex plane ...
0
votes
1answer
218 views

Does an analytic function maps a simple connected region into a simple connected region?

Suppose $f$ is analytic, say, in $\mathbb{C}$, and suppose $\Omega$ is a bounded simple connected open domain whose boundary we denote as $\Gamma$, then is $f(\Omega)$ also a simple connected domain ...
0
votes
4answers
162 views

Show: $\mathbb{C}=\overline{\mathbb{C}\setminus\left\{0\right\}}$

I have to show that $\mathbb{C}=\overline{\mathbb{C}\setminus\left\{0\right\}}$, what is very probably an easy task; nevertheless I have some problems. In words this means: $\mathbb{C}$ ...
3
votes
3answers
194 views

Möbius transform calculation, over an annulus

I started learning about Möbius transformations in my Complex Analysis textbook. This question appeared as an exercise (no solutions are provided, sadly): Let's say you have a Möbius transform that ...
4
votes
1answer
151 views

A step in the proof of the Riemann Mapping Theorem

Let $\Omega \subsetneq \Bbb C$ be open and simply connected. Let $\overline{\Bbb C}$ denote the Riemann sphere and assume without loss of generality that $0 \in \overline{\Bbb C} \backslash \Omega$. ...
4
votes
1answer
252 views

Big Rudin 1.40: Open Set is a countable union of closed disks?

Reading through Big Rudin, I have come across the following statement in the proof of Theorem 1.40: Let $S \subset \mathbb{C}$ be a closed set [in the topology induced by the complex modulus]. ...