Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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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 ...
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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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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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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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34 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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75 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) ...
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620 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 ...
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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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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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57 views

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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439 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
36 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 ...
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2answers
186 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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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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54 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 ...
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2answers
102 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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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 ...
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114 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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47 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 ...
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2answers
54 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 ...
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80 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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17 views

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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63 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 ...
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149 views

Totally disconnected vs. totally separated.

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 ...
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35 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 ...
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868 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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128 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.
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86 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 ...
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67 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, ...
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195 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 $(b_n)$?
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61 views

Metrics (Distances) on $\mathbb{F}$ Theorem Proof

I had a question regarding a Theorem I had come across that described metrics (distances) on ordered field $\mathbb{F}.$ Here it is: Theorem: If $\mathbb{F}$ is an ordered field, then $d(x,y)=|x-y|$ ...
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Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$? [closed]

Please how to prove that $d(x,y)=|x-y|^2$ is a distance on $\mathbb{R}$, I don't know how to solve the triangular inequality. Thank you.
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1answer
60 views

Linear bound on angles in an euclidean triangle.

I am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...
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337 views

Topology; Definition of the open ball and open sets confuses me

I just picked up T.W Gamelin’s book on topology. I started reading and got confused when I came to the definition of an open ball on the second page. It says $B(x;r) =$ All $y$ in the set $X$ such ...
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2answers
66 views

Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
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1answer
49 views

Subbases and half-planes

If $(X,d)$ is a metric space, it's easy to show that $H(x,y)=\{w\in X\mid d(x,w)>d(y,w)\}$ is open in the topology $\tau$ induced by $d$. Is, in general, $\{H(x,y)\mid (x,y)\in X\times X, x\neq ...
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At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state: Every compact metric space of covering dimension $n$ can be embedded ...
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Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm?

Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm? Suppose that $\lambda _n \to \lambda $, $\mu _n \to \mu ...
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91 views

How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
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44 views

A question about a perfect space and a linear order on it

Suppose I have a nonempty perfect Polish space $X$, and there's some linear order $<$ on it (it is not related to the topology on $X$ in any way). How can I prove that there is a point $y$ in $X$ ...
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91 views

Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I ...
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102 views

A geodesic metric space is a manifold on its own right. What are conditions for a Finsler space to be a manifold?

A geodesic metric space can locally be approximated by a vector space. This approximation provides it with a natural manifold structure. It means that geodesic metric space is more fundamental concept ...
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61 views

Is this a metric on matrices?

In the set of $n$-by-$n$ reversible real matrices, decide whether $$d(A,B)=\ln (\lVert A^{-1}B\rVert\cdot\lVert B^{-1}A\rVert)$$ defines a metric and/or semi-metric. Can you please help me to solve ...
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461 views

Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
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1answer
79 views

Continuity of evaluation maps in the topology of compact convergence on $C([0,\infty),\mathbb{R}^{n})$

I'm trying to prove that the evaluation maps $e_{x}:C([0,\infty),\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ given by $e_{x}(f):=f(x)$ are Lipschitz-continuous with respect to the metric ...
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concepts which is present in metric space but not in topological space

I want to know some concepts which is present in metric space but not in topological space. The one that first comes to mind is uniform continuity, equicontinuity i.e. concepts defined with some kind ...
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2answers
79 views

A problem similar to Banach fixed point theorem

a) Let $(X,d)$ be a complete metric space and let $T: X \to X$. Prove that if there exists a natural $n$ such that $T^n(x)$ (composition of $T$ $n$ times) is a contraction then $T(x)$ has a unique ...
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2answers
56 views

Prove ${x:d(x,p) < d(x,q)}$ is open in metric space $X$

$X$ is a metric space and $p \neq q$ $\in X$. I want to prove that $E=$ $\{x:d(x,p) < d(x,q) \}$ is open in metric space $X$. I think I can directly prove this by showing every point $x \in E$ ...
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1answer
57 views

Metric Spaces: closure of a set is the set of all limits of sequences in that set

I am studying metric spaces and got confused about many different ways of defining the closure. Let $S$ be a subset of $M.$ Then, the closure of $S$ is $ \{x \in M : \forall \epsilon>0, \ \ ...