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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Metric on function space for Pointwise Convergence

We say $\{f_n:[a,b]\to \mathbb{R}\}$ is a sequence of functions converging pointwise to $f:[a,b]\to \mathbb{R}$ if for any $x$ in $[a,b]$ we have $f_n(x)\rightarrow f(x)$. In this definition we ...
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3answers
28 views

Area of set-difference

Let $X$ and $Y$ be two open sets in $\mathbb{R}^2$, with $X\subsetneq Y$. Is it possible that $\text{Area}(Y\setminus X)=0$? Is it possible that $\text{Area}(Y\setminus Closure[X])=0$?
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1answer
18 views

poin to set distance lemma, whats the proof? [closed]

The lemma goes like this: Let $A \subset X$ be any subset of a metric space $(X,d)$ and let $x,y$ be any two points in $X$. Then proof that $$|D(x,A)-D(y,A)| \le d(x,y)$$ Where $D(x,A)=\inf [ ...
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1answer
26 views

uniform continuity of a function in a metric space

Let $A\neq\emptyset$ be a given subset of a metric space $(X,d).$ If $f(x)=d(x,A)$ show that $f$ is uniformly continuous on $X$.
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1answer
27 views

Prove uniform continuity of function

I was given $f: <1,+\infty>\times<1,+\infty>\rightarrow <0,+\infty> $ defined with $f(x,y)=\ln x+\ln y$ and metric on both spaces is induced by taxicab norm. I need to prove this ...
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0answers
16 views

Proving compactness in a geometric scenario

Let $C$ be a compact subset of $R^2$. Let $D$ be the set of all pairs of points $(P,Q)$ from $C$, such that the open segment between $P$ and $Q$ is contained in $C$: $$D = \{(P,Q)|P\in C, Q\in C, ...
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1answer
38 views

Proof with compact metric space and connectedness [closed]

Prove that compact metric space $(X,d)$ is connected if and only if for each pair of points $a,b \in X$ and $ \epsilon >0$ exist points $a_1,...,a_n \in X$ that $a_1 = a, a_n=b$ and ...
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1answer
16 views

distance-measure method to measure the distance between two matrixes(probability distribution)

I should find a suitable distance-measure method to measure the distance between two matrixes. The elements of such matrix is 0 to 1, and the sum of the all element is 1, so I think I could treat it ...
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0answers
17 views

Proof of Kuratowski-Wojdyslawski theorem

I was reading the Wikipedia page on Kuratowski Embedding, and the following result is stated: The Kuratowski–Wojdysławski theorem states that every bounded metric space $X$ is isometric to a ...
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1answer
31 views

Sequence and Series doubt

Suppose $x_n \to x$ in metric space $X$ and $y_n \to y$ in metric space $Y$. When can we say $(x_n,y_n) \to (x,y)$ ? i.e. what product metric will make it happen ?
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1answer
106 views

For what parameters does a sequence converge in $S$

Let $S$ be space of rapidly decreasing functions $f\in C_0^\infty(\mathbb R^n)$, that for any multi-indices $\alpha$ and $\beta$ there is a constant $M_{\alpha,\beta}$ such that $$|x^\alpha D^\beta ...
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1answer
30 views

Adherent values for a sequence in a metric space

I have these two definitions for an adherent value of a sequence the first is : $a$ is a an adherent value for $(x_n)$ iff $$\displaystyle \forall \varepsilon>0,\forall n\in \mathbb{N},\exists ...
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1answer
59 views

A problem in A Course in Point Set Topology by Conway, union of totally bounded sets

This is stated as a problem in A Course in Point Set Topology book by J. Conway: Let $\{E_n\}$ be a sequence of totally bounded sets. If $\operatorname{diam}E_n\to 0$ as $n\to\infty$, show that ...
2
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2answers
29 views

Negating the definition of a limit point

Below is a definition of a limit point: $E$ is a subset of a metric space $X$. $p \in X$ is a limit point of $E$ exactly when every ball around $p$ has an element $q \in E$ such that $q \neq p$. ...
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1answer
108 views

Jingle River (Berbed wire) Metric Problem

I want to prove that Jungle River metric is indeed a metric space, and determine it is open and closed balls. Firstly, i know that the metric is given by $x,y\in \mathbb{R}^2$, such that $x=(x_1,x_2), ...
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1answer
46 views

exercise about metric spaces: prove that the function is a metric

Let $(X,d)$ a metric space, $\alpha >0$ (fixed ) and $T: X \rightarrow X$ a map such that exist $n \in N$, where : $$ d(T^n x , T^n y) \leq \alpha^n d(x,y), \forall x,y \in X$$ Define $h(x,y) ...
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1answer
70 views

Prove that in $\mathbb{R}^n$, an open set is a countable union of closed balls. [closed]

If $A\subset \Bbb R^n$ is a open set. Then $A$ is a countable union of $\textbf{closed balls}$. Please, any reference to the previous theorem. Thank you all.
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1answer
37 views

How to compute antilogarithmic and superlogarithmic spaced values?

Let's suppose I have a range, e.g. $[100, 900]$. I want to compute 8 logarithmic spaced values $x_i={100, ..., 900}$. I use the following formula: $$x_1=\log(S)+\frac{(i-1)\log(S/L)}{n-1}$$ In the ...
3
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1answer
41 views

Simplexes in $\mathbb R^n$ have at most $n+1$ points

This is an exercise from the book Espaços Métricos (metric spaces) by Elon Lima. I'm translating it (the part of it that I'm having trouble with): Show that if $X\subset\mathbb R^n$ is such that ...
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0answers
26 views

Suppose $f \in B_{d_1}(g,\epsilon)$ can we conclude that $f \in B_{d_2}(g,\epsilon)$

Suppose $f \in B_{d_1}(g,\epsilon)$ can we conclude that $f \in B_{d_2}(g,\epsilon)$ where the space is $C[0,1]$ and $d_1$ is the metric induced by the $1$ norm and $d_2$ is the metric induced by the ...
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1answer
7 views

balls have empty boundary with regard to the $p$-adic norm

Let $p$ be prime, $a\in\mathbb{Q}$ and $r\geq0$. How can I show that the closed ball $D(a,r)$ in $(\mathbb{Q},|\cdot|_p)$ must have an empty boundary (with regard to the topology induced by the ...
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1answer
18 views

Is $T:(x,y)\mapsto(x+\alpha, y+x)$ mod $1$, expansive on $\mathbb{R}^2 / \mathbb{Z}^2$?

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha, x+y\right) \mod 1 $$ One ...
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1answer
38 views

Show that $S= \{ \left(\frac{i}{k},\frac{j}{nk} \right) : 0 \leq i < k, 0 \leq j < nk \} $ is an $(n,\epsilon)$-spanning set

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha \mod 1, x+y \mod 1 \right) ...
2
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1answer
159 views

How to show that the spherical metric satisfies the triangle inequality?

For $x,y\in \mathbb R^n$ define $$d(x,y)={\|x-y\| \over \sqrt{1+\|x\|^2} \sqrt{1+\|y\|^2}}$$ Here $\|x\|$ is the euclidean norm of a vector. How to prove that $d$ (the spherical metric) is indeed a ...
4
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2answers
68 views

Are these graphs all bipartite?

Given a number $D >0$, define a graph $G_D$ as follows. The vertices of $G_D$ correspond to points in the two-dimensional integer lattice $\mathbb{Z} \times \mathbb{Z}$. A pair of vertices $\{ ...
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3answers
74 views

Is being a Cauchy sequence equivalent to $ \lim_{n\to+\infty}d(x_{n+k},x_n)=0$ for every $k$?

Is this statement true? In a metric spase $(E,d)$, a sequence $(x_n)$ is Cauchy if and only if $ \forall k\in \mathbb{N}, \lim_{n\rightarrow+\infty}d(x_{n+k},x_n)=0$ I proved that ...
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1answer
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Every point of an open ball is a centre for the open ball.

Suppose $X$ is a nonempty set and $d$ is an ultrametric on $X$ i.e.,$$d(x,y)\le\max\{d(x,z),d(z,y)\}$$ for all $x,y \in X$. Suppose B is an open ball of $(X,d)$. Show that every point of B is a ...
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3answers
88 views

Bounded sequence has no convergent subsequence

How can you prove that in the metric space $(\mathbb{R},d)$ where $d(x,y)=|\arctan{x}-\arctan{y}|$ the sequence $(x_n)=n$ is bounded but it has no convergent subsequence ? Edit 1. Can I say that ...
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1answer
18 views

Application of inverse function theorem for several variable functions

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be continuously differentiable, and assume $Df(x)$ is invertible for all $x\in \mathbb{R}^2$. Also for any compact $K$ in $\mathbb{R}^2$, $f^{-1}(K)$ is ...
2
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2answers
88 views

Continuous function on complete bounded metric space need not be bounded

I came across the following old qual problem: Suppose $(X,d)$ is a complete metric space with finite diameter. Is every continuous function on $X$ bounded? It seems like the function $1/x$ on ...
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1answer
66 views

Propreties of sequences in a metric space [closed]

Please i have to study the exactenes of these assertions: Let $(x_n)_n$ be a sequence in a metric space $(E,d)$ 1)$(x_n)$ have no convergent subsequences 2)If $(x_n)_n$ is bounded then it has a ...
2
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2answers
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why is the following not a metric on $R^2$?

why is the following not a metric on $R^2$? All 3 conditions are getting satisfied $d((x,y),(x^{'},y^{'}))=|x|+|y|+|x^{'}|+|y^{'}|$ after many attempts by two of my friends please find the problem ...
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0answers
29 views

Comparison Triangle Angles

I am recently studying CAT(0) spaces and I have some doubts. (this because my intuition goes against what I wrote in classroom, thus I am not sure if I am doing something wrong or I ...
4
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2answers
63 views

Closed subsets in metric space

I want to prove that any closed subset $F$ from a metric space $(E,d)$ can be written as a denumerable intersection of open sets i.e., $$F=\bigcap_{n\in\mathbb{N}} \Theta_n; \Theta_n=\bigcup_{x\in ...
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0answers
25 views

Set similarity metrics

I have two sets $A$ and $B$. Each of them has a set of instances. Say $A = {a,b,c,d}$ and $B = {b,f,g}$. What are the most performing metrics out there to compute the similarity between the two ...
4
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0answers
48 views

For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
2
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1answer
51 views

To Prove that $(X,d)$ is not separable.

Let $X$ be the set of sequences in $[-1,1]$. Define $d$ on $X\times X$ to be $d(a,b) := \sup\{|a_n -b_n| :n \in \mathbb N\}$. Then $d$ is a metric on $X$; and show that $(X,d)$ is not separable. I am ...
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1answer
34 views

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?
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Proving intersection of dense subsets of a metric space X is the isolated points of X.

Suppose X is a metric space. Let $\mathscr C$ denote the collection of all dense subsets of X. Show that $\bigcap\mathscr C $ = iso(X). Thus the question asks to prove that every dense subset of X ...
2
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2answers
43 views

Property of distance and adherence

Please how to prove that in a metric space $(E,d),$ for $A,B\subseteq E$ that $\forall x\in E, d(x,A)=d(x,\overline{A})$ and that $d(A,B)=d(A,\overline{B})=d(\overline{A},\overline{B})$ and ...
2
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0answers
36 views

Gelfand width of $l_1$ unit ball in $\infty$ metric

The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not ...
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Equivalence of Theorems; the sphere on $\ell^2$ is finitely oscillation stable.

I just started reading the book Dynamics of Infinite-dimensional Groups, by Pestov, and right in the introduction the following theorem by Milman is cited: Let $\mathbb{S}^\infty$ denote the sphere in ...
3
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1answer
41 views

Gradient of the distance function

Let $\Omega$ be open, bounded subset of $\mathbb{R}^n$. Let $d(x):=dist(x,\partial\Omega)$ denotes the distance of the point $x\in\Omega$ from the boundary $\partial\Omega$. Define function ...
5
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1answer
60 views

Totally disconnected space in which some quasicomponents have interior?

Assume all spaces are metric. Question. Does there exist a space $X$ which is totally disconnected (the components of $X$ are singletons), yet some quasicomponent of $X$ has nonempty interior? I ...
2
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1answer
34 views

Elements with infinite roots in p-adic

Let $\mathbb{Q}_p$ the $p$-adic completion of $\mathbb{Q}$ and $$S=\{x\in\mathbb{Q}_p:1+x\mbox{ has n-th root in }\mathbb{Q}_p\mbox{ for infinite }n\in\mathbb{N}\}$$ I have to show that ...
2
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3answers
298 views

A not complete metric space?

Please how to prove that the space $\mathbb{R}$ endowed with the metric $d(x,y)=|e^x-e^y|$ is not a complete space? I don't find a Cauchy sequence but not convergent Please Thank you.
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4answers
52 views

How is a metric space a topological space? [duplicate]

I learned about metric spaces and topological spaces but I don't see how they correlate. How does a metric space follow the properties of a topological space.
2
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2answers
53 views

Prove that a set in a metric space cannot be both open and closed.

If I have a metric space $X$, and $E \subset X, E \ne X, E \ne\emptyset$. I want to prove that E cannot be both open and closed. I have two strategies, but I am not able to finish them: I assume ...
0
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1answer
52 views

Convergence of distances in metric space

If $(X,d)$ is a metric space, $(x_n)$ and $(y_n)$ are Cauchy sequences in $(X,d)$. How do i show that $(a_n):=d(x_n,y_n)$ converges? Here is what i did: Let $(x_n)$ and $(y_n)$ be Cauchy sequences, ...
1
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4answers
151 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?