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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In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
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Lebesgue prolongation of a measure and metric spaces

Anyone knows the connection between these two things. Tha books talks about the measure $m^*$ in a ring $\mathcal{R(G_m)}$; $m^*(A \triangle B)$ can be seen like a distance between two sets of the ...
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152 views

Let $X$ be a metric space with metric $d$. Show that $d:X \times X \longrightarrow \mathbb{R}$ is continuous.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I want help with my proof in particular. Let $X$ be a metric ...
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1answer
90 views

Cauchy sequence in compact metric space

Suppose $f:X\rightarrow X$ continuous function, $X$ is compact metric space with $\rho(f(x),f(y))<\rho(x,y)$ for any $x\neq y$. Let $x_n=f(x_{n-1})$, with $x_0\in X$ arbitrary. I want to show that ...
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Does “uniformly isolated” imply closed?

Let $X$ denote a complete metric space and consider a subset $A \subseteq X$. Call $A$ uniformly isolated iff there exists $r > 0$ such that for all $a \in A$, we have that $B_r(a) \cap A = \{a\}$. ...
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amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
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1answer
29 views

Homeomorphism of Interior of Convex Polygon to Open Unit Disk

Show that the interior of a non-degenerate convex polygon in $\mathbb{R}^2$ is homeomorphic to the open unit disk in $\mathbb{R}^2$. My attempt: Let $P$ be the set of points in the polygon, let ...
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18 views

Two forms of application of metric tensor to get differential length

I'm reading the monograph "Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity" by Joseph Kolecki, now retired, of NASA. I have a ...
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1answer
68 views

Is there any proof for this simple observation? [duplicate]

If we consider the Euclidean space $R^3$, it is simply the space where we live. Here we can find only four point such that distance between any two points is a constant. If we consider the Euclidean ...
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1answer
59 views

Is a Riemannian metric a $2$-form?

In Lee's Riemannian Manifolds; An introduction to Curvature, he defines a Riemannian metric as an element of $\Gamma(T^2_0M)$, a $(2,0)$-tensor. Is this the same thing as a $2$-form? Is there a ...
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Compact sets and Kuratowski limit

I have been struggling with the following claim: Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the ...
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1answer
37 views

Discontinuous function whose restriction on closed sets is continuous

Let $X$ a metric space, $\{U_i\}$ a collection of non-empty closed sets whose union is all of $X$. Give an example of a function $f:X\rightarrow \mathbb{R}$ such that the restriction $f|_{U_i}$ is ...
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Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
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3answers
103 views

Compactness under different metric?

Consider the metric $\rho(x,y)=\frac{|x-y|}{1+|x-y|}$ on $\mathbb{R}$. Is $(\mathbb{R},\rho)$ compact? In order to show that is not, I wanted to find a sequence such that any subsequence is ...
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2answers
60 views

If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?

Let $X$ denote a metric space. Whenever $S \subseteq X$ and $r \in \mathbb{R}_{\geq 0}$, write $f(S,r)$ for the following set. $$\{x \in X \mid \exists s \in S : d(x,s) \leq r\}$$ Question. ...
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1answer
44 views

Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
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1answer
22 views

Find continuous $f$ s.t. $f(x,y)^2 = x^2$ ($f: H \to \mathbb{R}$, H connected)

(i) Show that a metric space M is connected if and only if every continuous integer-valued function on M is constant. (ii) Show that $H = \{(x, y) \in R^2 : x > 0\}$ is connected. By ...
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1answer
45 views

Proving there exist convergent subsequences for bounded sequence of real numbers

I'm trying to teach myself some basic topology by self-studying from Intro to topology by Mendelson. I'm stuck on one of the exercises and can't figure out how to proceed with the proof. The question ...
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1answer
34 views

Prove that square metric on $\mathbb{R}$ is in fact a metric.

In Munkres's topology, he proves that square metric on $\mathbb{R}^n$ is in fact a metric. By the square metric, I mean this function: $P:\mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{R}$ ...
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Easier proof of “countable hypocompactness”

I am interested in the following result, which appears as an old qual problem: Let $X$ be a metric space and $\{U_i\}$ a countable open cover. Prove that there exists a countable open refinement ...
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2answers
50 views

Geometry of Metric Spaces

I'm reading a book on Metric Spaces and the author is always talking about the "geometry" of some metric spaces, but he doesn't say what he means by geometry. For example: Despite the fact that ...
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1answer
28 views

Why is this set closed?[metric-spaces]

I am reading a note, where part of it is this: Why is S' closed? I have tried to argument like this, but I am not able to finish the argument: Let $\{x_n\}$ be a convergent sequence in S', then ...
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3answers
43 views

Prove the equation $\vert d(x,z)-d(y,t)\vert\leq d(x,y)+d(z,t)$

I know how it verified the following equation: $$\vert d(x,y)-d(x,z)\vert\leq d(y,z)$$ where $x,y,z$ is arbitrary points of metric space $(X, d)$ But I didn't now how to prove the follow equation: ...
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89 views

Why in open balls is radius $r>0$?

the usual definition is the following: Def.1: let be $(a,f)$ a metric space, $c \in a$ and $r \in \Bbb{R}_{>0}$, the open ball of radius $r$ about $c$ is the set $$\mathcal{B}_f(c,r)=\{x \in a| ...
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Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
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29 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
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43 views

Gromov-Hausdorff distance between a “Line segment” and a “Zylinder”

I want to prove the following statement: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = \frac{\pi}{2} $ where $d_{GH}$ denotes the Gromov-Hausdorff ...
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78 views

Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
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293 views

Given an example of a metric space in which every sphere has two centers

This is a question from Wilansky "Topology for analysis", P.15 Prob. 103 Maybe I was thinking too Euclidean, I can't come up some other "centers" of the sphere :(
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4answers
196 views

The set of points where two continuous functions agree is closed.

I want to prove that if $f,g$ are continuous functions from a topological space $(X,\tau)$ to a metric space $(Y,d)$ then the set $$ A = \{ x \in X : f(x) = g(x) \} $$ is closed. I found a very ...
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Discrete Fréchet distance vs Dynamic Time Warping

Is Dynamic Time Warping the same as using the Discrete Fréchet Distance with a reparameterization so that the sum of point-to-point distances is minimized (instead of the maximum as usual)?
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Sketching the unit ball centered at the origin of the metric $d(x,y)=\vert x_1 -y_1 \vert + \vert x_2-y_2 \vert$ in $\mathbb{R}^2$

I am having some diffucilty sketching the unit ball centered at $(0,0)$ for the metric given by $$d(x,y)=\sum_{i=1}^n \vert x_i -y_i \vert$$ in $\mathbb{R}^n$ for $n=2$. If $n=2,$ the unit ball is the ...
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213 views

How do I turn my verbal argument into something formal in [Real Analysis]? (proving every compact set is bounded)

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
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36 views

How do I turn my verbal argument into something formal in Real Analysis? [duplicate]

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
3
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22 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
3
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1answer
51 views

Can all null-homotopy be made differentiable on arbitrary metric space?

Let $M$ be a metric, and assume that it is simply connected. For a closed curve $f$, we define it to be differentiable iff for any $x$ then $\lim\limits_{h\rightarrow 0}\frac{d(f(x),f(x+h))}{h}$ ...
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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 ...
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1answer
27 views

Proving that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance in $\mathbb{R}^2$

I was asked to prove that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance function in $\mathbb{R}^2$. I've got myself stuck with proving the triangle inequality. Can someone give me an hint ...
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1answer
50 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
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$ max \{x : 0 \leq x < 1\} = ? $

As per the title, what is the maximum value: $$ \max \, \{x : x \in \mathbb R, 0 \leq x < 1\} = ? $$ This question came to me when considering the supremum metric applied to the set of functions ...
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1answer
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CAT(0) space is geodesic

Proving that being CAT(0) implies being geodesic, my professor used the following fact: If for every pair of points $x$ and $y$ in a complete metric space $X$ there exists a point $m \in X$ such that ...
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A metric on $\mathbb{N}$

Define a metric on $\mathbb{N}$ by fixing a prime, $p$, and setting $$d(x,y)=\begin{cases} 0 & x=y \\ p^{-k} & \text{otherwise} \end{cases}$$ where $p^k$ is the highest power of $p$ that ...
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2answers
37 views

Number of open sets in a metric space

I have got the following question which I could not solve: can a metric space have exactly 36 open sets? I believe if the metric space is finie, then it has to be discrete and so the number of open ...
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1answer
52 views

How one shows that the triangle inequality holds for this metric?

Define $d:\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{R}$ by $\displaystyle d(m,n)=\frac{1}{\sup\{l\in\mathbb{N}: l!\mid\lvert m-n\rvert\}}$ with the obvious interpretation that when the supremum ...
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7answers
150 views

Convergence in a metric space

Is it possible to define a metric on $\mathbb R$ such that $(1,0,1,0,...)$ converges on $(\mathbb R, d)$? I believe it is impossible. But how to show analytically? Any hint would be appreciated.
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Completeness of Locally Compact Metric Space and Group of Isometries

Let $X$ be a locally compact metric space, and suppose that the group of isometries of $X$ acts transitively. Show that $X$ is complete. (This is 2nd part of a problem. In first part I showed that for ...
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88 views

Realizing $\mathbb{Z}/n \mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ as an isometry group

Every finite group arises as the isometry group of a subspace of an euclidean space (Albertsona, Boutin, Realizing Finite Groups in Euclidean Space). What are natural examples of spaces realizing the ...
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1answer
24 views

Lipschitz distance

The Lipschitz distance between two metric spaces is defined by $$d_{\mathcal L}(X, Y) = \inf_f \log(\{\max\{\text{dil}(f), \text{dil}(f^{-1})\})$$ where $$\text{dil}(f) = \sup_{x, y \in X} \frac{ ...
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4answers
146 views

How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...
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1answer
30 views

Isometry fixing an open set pointwise is identity

Let $X$ be a metric space and $F$ an isometry of $X$. Suppose $F$ fixes each point of a non-empty open set $U\subset X$. Under what conditions on $X$ does it always follow that $F=\mathrm{id}$? I ...