1
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2answers
31 views

Does a bounded continuous function map Cauchy sequences to Cauchy sequences?

I only ever see the example of $f:(0,1]\rightarrow \mathbb{R}$ where $f(x)=\frac{1}{x}$as that of a continuous function that does not map Cauchy sequences to Cauchy sequences. Are there examples of ...
1
vote
1answer
63 views

Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
1
vote
2answers
27 views

Proving compactness of the extended complex plane

Prove that $(\overline{\mathbb C}, \overline{d})$ with $\overline{d}(z,z')=d(\phi(z),\phi(z'))$, where $d$ denotes the euclidean distance in $\mathbb R^3$ and $\phi$ is the inverse of the ...
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 ...
0
votes
1answer
47 views

Show that $\ c_X(p,q) \le d_X(p,q)$, for $ p, q \in X$

Update I'm trying to show the Corollary, but I have stuck...That is: For any complex space $X$, we have: $$\begin{align} (1).\ c_X(p,q) &\le d_X(p,q),\ \text{for}\ p, q \in X \\ (2).\ ...
0
votes
1answer
42 views

Closed sets and sequences in Metric spaces

Suppose $x \subset X$ is a closed set, the sequence {$ {x_j}$}${ } \subset F$ and $x \in X$. Show that if $x_j \to x$ as $j \to \infty$, then $x \in F$ Okay so I really don't know where to start with ...
0
votes
0answers
31 views

Reality condition on a metric

I'm studying a coördinate-transformation on a 2n-dimensional real manifold, where I can locally define the coördinates as $(x^1,...,x^{2n})\in\mathbb{R}^{2n}$, and transform them 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 ...
0
votes
1answer
40 views

Proving that a metric on space of analytic functions is equivalent to compact convergence

Let $U\subseteq \mathbb C$ be open and $\mathscr A(U)$ consist of all analytic functions on $U$. I can easily prove that there exists a sequence $K_n$ of compact sets in $U$ so that ...
3
votes
1answer
81 views

Show that $d(z,a)=|\frac{z-a}{1-\overline{a}z}|$ is a metric on the unit disk

For any $z, a$ in the open unit disk, we define$$d(z,a)=\left \vert \frac{z-a}{1-\overline{a}z} \right \vert$$ Show that $d$ is a metric on the open unit disk. It is easy to show ...
1
vote
2answers
71 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
55 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
74 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 ...
4
votes
1answer
288 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]. ...
0
votes
1answer
104 views

Fréchet mean of the spherical shape space

The Fréchet mean of a general subspace is defined as $$F(x)=\int_M dist(x,y)^2d\mu(y),$$ where $\mu$ is the probability measure on a general metric space $(M,dist)$. I think the sample mean of ...
1
vote
0answers
64 views

Fréchet mean for a general shape space

I am posting this question in order to gain a better understand of what the Fréchet mean is for a generalised shape space. So firstly I gather that the Fréchet mean of a probabilty measure $\mu$ on a ...
0
votes
1answer
54 views

The Differentiability of $f$ in $\mathbb R^2$

$f: D(\subset \mathbb C)\to \mathbb C$ is differentiable & real-valued ($D$ being a domain) $\Rightarrow$ $f$ is constant in $D$. Also the usual metrics on $D$ & $\mathbb R^2$ are identical. ...
3
votes
0answers
78 views

The canonical (1,1)-form on a compact Riemann surface gives locally a subharmonic function

Let $X$ be a compact connected Riemann surface of genus $g>0$. We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ ...
0
votes
2answers
111 views

Is it easy to see that this function is subharmonic?

I would like to know if the function $$\frac{1}{(1+\vert z\vert^2)^2}$$ is subharmonic on $\mathbf{C}=\mathbf{P}^1-\{\infty\}$. Motivation. The Fubini-Study metric on the complex projective line ...