Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Show that the mobius transformation, $M:H \rightarrow H$ is a homeomorphism

Where $M$ is defined: $M(z) = \frac{ez+f}{gz+h}$ And, $H = \{z= x+iy \in \mathbb{C} \space| x,y \in \mathbb{R}, y > 0 \}$ is the upper half plane, with the induced topolgy such that the ...
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Q: Nowhere dense sets.

Given $X$ a metric space, $A\subset X$ a nowhere dense set. Show that every open ball $B$ contains another open ball $B_1 \subset B$ such that $B_1 \cap A = \emptyset$. EDIT: I modify my proof ...
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Compact subset of an open set in the complex plane

I would just like to ask a question. Suppose $K$ is a compact subset of an open set $V$ in the complex plane, how would you prove that there exists an $r > 0$ such that the union $E$ of the closed ...
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question in metric space completion properties

Dear all I was given this question in topology which I would really appreciate help with: I am asked to prove that for every metric space we have that the space itself is totally bounded if and only ...
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Show that a countable dense subset $D \subset X$ is not a $G_{\delta}$

Given $X$ a complete metric space with no isolated points and $D \subset X$ a countable dense subspace, show that $D$ is not a $G_{\delta}$. I am quite lost in trying to use the hypothesis of the ...
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Euclidean distance on R and Q

I have to answer the following three questions. Given $\epsilon > 0$, a metric space $(V,d)$ and a element $a \in V$ i) Show that for all $x \in B(a;\epsilon)$ there exists a $n \in \mathrm{N}$ ...
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31 views

Is there a name for this property of a subset of a metric space?

Our 'intro to Analysis' professor intruduced compactness of subsets of metric spaces by first introding what he calls 'rijcompact' (sequential compactness), and going on to prove that it is equivalent ...
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Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
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178 views

In a metric space, is every convergent sequence bounded?

In $\mathbb{R}$ and $\mathbb{R}^p$, this is true, but is it true in every metric space? I suppose not, but what other condition would I have to put on the metric space in order for it to have this ...
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Question regarding metric spaces and open balls

I have just started with metric spaces and am reading Karl Stromberg's Introduction to Real Analysis. In one of the examples, the following is stated: If $X=\mathbb N\subset \mathbb R$ with the ...
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49 views

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces Ok so I know I have to use Baire's Cathegory Theorem here. And I've done the following, lets suppose on the ...
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48 views

The metric space and the balls.

Let $M$ be a metric space with a distance function $d$, and let $a,b,c\in M$ be given. Let $r,s,t>0$ and assume that $c\in K(b,s):=\lbrace x\in M\mid d(b,x)<s\rbrace$. a) Show that if ...
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26 views

A simple version of the Picard-Lindelöf Theorem

I wish to ask a particular question about following the proof in this theorem, and thought the best place to come might be here. It is as follows: First, we have a differential equation that ...
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60 views

Existence of a metric space where each open ball is closed and has a limit point

Show that there exists a metric space in which every open ball is closed and contains a limit point. I think that the space $\{\frac{1}{n}\mid n\in\mathbb{N},n>0\}\cup \{0\}$ with the standard ...
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1answer
33 views

Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$

Let $A,B$ be two compact subsets of $X$ where $(X,d)$ is a metric space. 1.Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$ where $d(A,B)=\sup\{d(a,b):a \in A;b\in B\}$ 2.Show that ...
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Munkres Section 20 Exercise 3b. Proof verification.

Let $X$ be a metric space with metric $d$. (a) Show that $d:X\times X\to\Bbb R$ is continuous. I've shown this already. (b) Let $X'$ denote a space having the same underlying set as $X$. Show that ...
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Prove $\tilde d: M\to \mathbb{R}; x\mapsto \tilde d(x) = d(x,A)$ is continous

Let $A$ be a non-empty subset of a certain metric space $M$. Prove that $\tilde d: M\to \mathbb{R}; x\mapsto \tilde d(x) = d(x,A)$ is continous. (where $d(x,a) = \inf\{d(x,a): a\in A\}$) ...
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56 views

Is it always true that the complement of a closed set is open?

In a metric space $M$, by definition, a subset $F$ is closed if $M\setminus F$ is open. However in a general topological space $T$, say with topology $\mathcal{T}$ is this always true? For example ...
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Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$

Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$ I have encountered this exercise and I don't know where to start. There is a lot of important ...
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How to obtain the metric tensor of a principal bundle total space given a connection (assuming it is metric compatible) in the total space?

The title says it all. In a principal bundle I know the connection defined in the total space. How can I calculate the metric that would be compatible with this connection.
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25 views

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete.

Show that $C_{0} = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}: a_n \rightarrow 0\}$ is complete. I've already seen that this question has been asked, and already answered, however, I've managed ...
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Prove that $X$ contains exactly two clopen sets if and only if every nonempty proper subset of $X$ has a nonempty boundary.

Let $X$ be a metric space. Prove that $X$ is connected if and only if every nonempty proper subset of $X$ has a nonempty boundary. proof: Suppose X is connected, then $X$ contains exactly two ...
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42 views

Show that $a\to f(a)$ from $A$ to $S$ is continuous.

I am reading "Continuity" in Metric Spaces Suppose $S\subset \mathbb R$ is a closed set. Suppose $A\subset \mathbb R$ has the property that for every $a\in A$ there is a unique nearest point $f(a)$ ...
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Contracting subsets

Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
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36 views

Metric spaces and fix point [closed]

I saw this problem in my course of Intr. to the topology: Let $(X,d)$ be a compact metric space and $$f :(X,d) \rightarrow (X,d)$$ a continuous function such that: $\quad d(f(x);f(y)) < ...
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Show that $B(X)=\{f:X \rightarrow \mathbb{R}: f \text{ is bounded}\}$ is complete.

Given a metric space $(X,d)$ consider the metric space $B(X)=\{f:X \rightarrow \mathbb{R}: f \text{ is bounded}\}$ with the distance $d_{\infty}(f,g)=sup_{x\in X}|f(x)-g(x)|$. Show that ...
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22 views

Is this proof involving complete metric spaces correct?

Show that if every closed ball of a metric space $(X, d)$ is complete then $ X$ is complete. I thought the following: given $(x_n)$ a Cauchy sequence in $X$, we have that the set $A= \{x_{1}, ...
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6 views

2 dimensional representations of v dimensional points based on distance

I have a set of points $x_1, \dots, x_n \in \mathbb R^v$ I have a measure of the distance between each one of these points $D \in \mathbb R^{n\times n}$ where $D_{i,j}= distance(x_i, x_j)$ I would ...
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35 views

$[0,1]\times\mathbb{N}/(0,k)$ not metrizable.

$X = [0,1]\times\mathbb{N}/((0,1)\sim(0,2)\sim\dots)$. I read that $X$ is not metrizable since sequence $\{(\frac{1}{n},n)\}$ is closed in $X$ and therefore does'n have limit. But i don't understand ...
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63 views

Some questions about écarts

An écart for a set $X$ is a non-negative real-valued function $e:X\times X\rightarrow \mathbb{R}$ such that $e(x,y)=0$ if and only if $x=y$; for each positive number $s$ there is a ...
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53 views

multiplying two metrics

Let $(X,d)$ and $(X,d^\prime)$ are metric spaces ,is $d×d^\prime$ metric on $X$ ?I try to prove triangle inequality , I write two triangle inequalities for $d $ and $ d^\prime$ but it not true.
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Proving $\ell^p$ is complete

Let be $1\leq p\in\mathbb{R}$, denote: $$\ell^p(\mathbb {R})=\left\{(x_n)\subset \mathbb{R}: (x_n) \mbox{ is a sequence with } \displaystyle\sum_{n=1}^{\infty}|x_n|^p<\infty \right\}$$ ...
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135 views

Is this a compact space?

Let $A=\{x:d_\infty(x,0)\le 1 \}$, the subspace of the space of bounded sequences $x=(x_n)^\infty_{n=1}$, $x_n\in \mathbb{R}$, with metric $\{x:d_\infty(x,y)= sup_n |x_n-y_n| \}$. The answer says it ...
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Conceptual problem regarding distance between two sets.

Given a metric space $(X,d)$ and two non empty subsets $A,B \subset X$ we define the distance between $A$ and $B$ as $$ d(A,B) = \inf\, \{d(a,b) : a\in A, b\in B\} $$ My question is the following: ...
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39 views

Is the distance function continuous? [closed]

Is the distance function continuous? I know that distance function is continuous, give an example of distance function that is not continuous.
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Example 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be open?

Let $N$ be the following subset of $\mathbb{R}^2$: $$N \colon= \{ \ (x,y) \in \mathbb{R}^2 \ \colon \ \vert x \vert < \frac{1}{y^2+1} \ \}.$$ Then intuitively it is apparent that $N$ is open. ...
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Characterization of a quotient space

Given the space $C^n[0,1]$ of all real functions of class $C^n$ in $[0,1]$, let $\tilde{d}^j := d_\infty(f^{(j)},g^{(j)})$ a pseudometric $(j=1,\dots,n)$ on $C^n[0,1]$. Here $f^{(j)}$ mean the ...
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1answer
35 views

Prove/disprove space is complete with metric defined by an integral (triangle inequality still missing in metric part)

I have a two part question: I need to show that $d(f,g)=\int_{-1}^1\! |f(x)-g(x)| \, \mathrm{d}x$ is a metric in $C((-1,1),\mathbb{R)}$ and furthermore prove/disprove that the space ...
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Equivalence of sequence spaces

Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i ...
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find mean of matrices $A_i, A_j$ given $d_{A_{ji}}=\ln{\left|\left| A_{ji} \right|\right| \left|\left| A_{ji}^{-1} \right|\right|}$

Given a finite set $\mathbb{A}$ of $k$ like-shaped, square, non-singular matrices $A_i\in\mathbb{R}^{n\times n}$, let's define $A_{ji}=A_j A_i^{-1}$, then the distance of the two matrices $A_i, A_j$ ...
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Understanding extremal Lipschitz functions

I am new to concept of extremal Lipschitz functions and I have several basic question I'm still unsure about. To fix notation let $(X,d)$ be a metric space, $Lip(X)$ Banach space of Lipschitz ...
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39 views

Prove that if $S\subset \mathbb{R}^n$ is not countable, then there exists $x \in S$ such that $x$ is a condensation point.

Let $S \subset \mathbb{R}^n$ with the usual metric. A point $x \in \mathbb{R}^n$ is said to be a condensation point of $S$ if for all $r>0$, $B(x,r)\cap S$ is not countable. Show that if $S$ is ...
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trouble in getting triangle inequality

Let $l_{2}$ be the set of all infinite sequences , $ (x_{n})$ such that $\sum_{n=1}^ {\infty} x_{n}$ converges. Define $$d(x,y)= \sqrt{\sum_{n=1}^{\infty} (x_{n}-y_{n})^{2}}$$ for each $x=(x_{n})$ ...
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34 views

Metrics on the set of natural numbers

I am trying to find a metric d on $ \mathbb{N} $ that is not equivalent to the discrete metric $ d_{\{0,1\}} $. Thus far I got a metric with the following properties: $ d(x, x_n) \in [0,1) \forall n ...
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33 views

Covering number definition, general metric space question.

While reading "On the mathematical foundations of learning" by F.Cucker and S.Smale I came across this definition: Let $S$ be a metric space and $s>0$. We define the covering number ...
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Topologically equivalent metric

Show that in $\mathbb{R}$ the distance $d'(x,y)=\left|\frac{x}{1+|x|} - \frac{y}{1+|y|} \right|$ is topologically equivalent to the usual metric in $\mathbb{R}$, $d(x,y)=|x-y|$ But ...
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every nonempty compact, locally path-connected and connected metric space is path-connected [duplicate]

I wanna prove that if $M$ is nonempty compact, locally path-connected and connected metric space then it is path connected. I think to prove this the best way is to show that between every to points ...
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2answers
75 views

Prove that $f^{-1} (F)$ is closed

A set $F \subset \mathbb R$ is closed if for any convergent sequence $\{x_n\}$ in F converges, we have $\lim_{n \to \infty} x_n=x \in F $. How to Prove that if $f :\mathbb R \to \mathbb R$ is ...
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1answer
33 views

Why is the open interval $(0, 1)$ a Polish space?

Wikipedia gives as an example for Polish spaces the open interval $(0, 1)$. Can somebody explain to me how $(0,1)$ can be Polish? $(0, 1)$ has to be metrizable so that it is complete, which means ...
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359 views

Express unit sphere as countable union of great circles?

Let $S = \{x\in \mathbb{R^3} | d(x,(0,0,0))=1\}.$ Is it possible that $S$ is a countable union of “great circles”? A great circle is the intersection of $S$ with a plane through $(0,0,0)$. What ...