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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86 views

A continuous function on the real line such that the preimage of every point is either empty of has exactly 3 points

Let $f : \mathbb{R} \to \mathbb{R}$ be a function with all fibres $(\lbrace{x \in \mathbb{R}| f(x) = c\rbrace} = f^{−1}(c)$, $c \in \mathbb{R})$ either empty or consisting of exactly three points. ...
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16 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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275 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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2answers
47 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. ...
2
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4answers
159 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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2answers
2k views

prove that the set of rational numbers is not connected on the real line [on hold]

Could someone help me through this problem? Prove that the set of rational numbers is not connected on the real line
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0answers
3 views

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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1answer
24 views

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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2answers
195 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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0answers
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 ...
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18 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 ...
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0answers
24 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 $r(x)$ ...
3
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0answers
27 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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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 ...
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1answer
25 views

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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2answers
116 views

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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1answer
24 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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50 views

$ 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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3answers
80 views

Coupling methods

Distance between probability measures Let $(X,d)$ be a compact metric space, and let $\mu$ and $\nu$ be two probability measures on $X$. We can define the Wasserstein distance between $\mu$ and $\nu$ ...
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1answer
44 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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0answers
21 views

triangle inequality in a metric space

`Let X=set of natural numbers.A metric "d" is defined on X such that d(m,n)= 0 , if m and n are same number = 1+ 1/(m+n) ,if m and n are different numbers. where m and n are natural ...
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2answers
34 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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7answers
148 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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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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2answers
337 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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0answers
12 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
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0answers
18 views

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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28 views

examples on metric space on set of natural numbers [closed]

Let $d$ be a function defined on $\mathbb N$, such that: $$d(m,n) =\begin{cases}0, & \text{if $m=n$} \\1+ \frac{1}{m+n}, & \text{otherwise} \\\end{cases}$$ where $m,n \in \mathbb N$. Show ...
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1answer
213 views

Non-empty intersection of open balls in $R^n$ contain open balls

I want to prove that if the intersection of two open balls about the points $x, y$ (resp.) is non-empty, then there exists a third ball centered at some point $z\in B_{\epsilon 1}(x)\cap B_{\epsilon ...
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4answers
138 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
19 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{ ...
2
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1answer
68 views

d' is finer than d on compact space $\implies$ $\forall \epsilon \ \ \exists \epsilon' \ \ \forall x \ B_{d'}(x,\epsilon') \subseteq B_d(x,\epsilon)$

This is my conjecture, but I guess I am missing the key idea for the proof (or my conjecture is wrong) Let d and d' be two metrics on a compact space $X$ ($X$ is compact with respect to both ...
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3answers
864 views

Difference between Norm and Distance

I'm now studying metric space. Here, I don't understand why definitions of distance and norm in euclidean space are repectively given in my book. I understand the difference between two concepts when ...
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1answer
27 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 ...
4
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3answers
386 views

Continuous functions between metric spaces are equal if they are equal on a dense subset

If two functions defined on metric spaces $X$ and $Y$ are equal on a dense subset of $X$ and are continuous also, then are they equal on all of the metric space $X$?
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1answer
425 views

Picturing Urysohn's Metrization Theorem and Urysohn's lemma?

In my Topological course we have this lemma. [Urysohn's lemma] Suppose that $X$ is a topological space. Then $X$ is normal if and only if, for each pair of disjoint closed subsets $A$ and $B$, there ...
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0answers
19 views

Finsler Metric from page 2 of the book by Chern and Shen.

Physicist here not a mathematician. I am trying to understand the notation for the Finsler metric in Chern and Shen's book. The equation is $$\textbf{g}_y(u,v):=\frac{1}{2} ...
2
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1answer
106 views

Length minimizing curves are geodesic segments

I have a metric space $(X,d)$, a geodesic arc is defined to be a continuous function $\gamma : [a,b] \rightarrow X$, $a < b$, which is (globally) distance preserving and geodesic segments are ...
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2answers
47 views

Show that for any subsets $A,B\subset X$: (i):$d(A\bigcup B)\leq d(A)+d(B)+d(A,B)$ and (ii) $d(\bar A)=d(A)$

Let $d$ be a metric on $X$. Show that for any subsets $A,B\subset X$: (i) $d(A\cup B)\leq d(A)+d(B)+d(A,B)$ (ii) $d(\bar A)=d(A)$ I found this hard to prove because the diameter ...
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50 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
2
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1answer
45 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
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1answer
23 views

Lipschitz continuous set-valued map

Let $X$ and $Y$ be metric spaces and $F:X\to2^Y$ be a set-valued map. Suppose that $2^Y$ is endowed with the Hausdorff metric. I wonder about sufficient conditions on $F$ that ensure this map is ...
2
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1answer
60 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
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2answers
25 views

Does a (local) uniqueness theorem exist for geodesics in metric spaces?

In Riemannian geometry, we have the next result. Let $M$ be a smooth manifold with an affine connection. For any point $p\in M$ and for any vector $v\in\mathrm T_pM$, there is a unique geodesic ...
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3answers
108 views

Proof involving a vacuously true statement

Let $S$ be a finite subset of a metric space. Show that it is closed. I know a set is closed if and only if it contains all of its accumulation points. Let $x$ be an accumulation point of $S$. I want ...
3
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1answer
40 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
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1answer
49 views

prove that there is maximum and minimum for C[0,1]

let $C[0,1]$ be the set of continuous functions $f:[0,1]\rightarrow\mathbb R$ and let $A\subset C[0,1]$ be the sub set of twice differentiable functions in $C[0,1]$ which satisfies: $\forall f\in ...
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57 views

We know that a Compact set is closed. However a finite discreet set is compact but not closed (contradicting the theorem?)

We know that a Compact set is closed. We also know that a finite discreet set is compact (as every cover has a finite sub cover). However a finite discreet set is not closed (contradicting the ...
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1answer
20 views

metric-spaces closure

The line in my writing symbolizes the closure, where we add all the boundary points. Is it true that: $\overline{A \cap B}=\overline{A}\cap {B}$? If B is closed, but we do not know if or if not A ...
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80 views

Metrizability under homeomorphism?

Is metrizability preserved under homeomorphism? That is, suppose that you have a topological space $(X, \tau_1)$ whose topology comes from a metric $d$, and you have another topological space $(Y, ...