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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Random Geometric Graph in unit disk

According to the Wikipedia article, to define a random geometric graph, one needs a metric space. In the examples they give for 2D random geometric graphs, they say that "an RGG can be constructed ...
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I'm not certain this makes any sense: Matrix Multiplication of Metric Tensor for calculating arclength

I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Arclength Where in it gives the euclidean measure of distance as $$ ds^2 = E du^2 + 2 F du dv + G dv^2 $$ Equivalently as $$ ds^2 ...
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Is it true that any continuous function $f$ on $[0,\infty)$ can be approximated by polynomials?

I think it's true, but how to prove it? By weierstrass approximation theorem we can approximate uniformly any continuous function on a closed interval $[a,b]$ by a sequence of polynomial function ...
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If $f$ is not uniformly continuous function on $(0,1)$ then $f$ can't be extended to a continuous function on $[0,1]$.

Is this proposition correct? If it is correct can anyone prove it? Because I need it to prove that if $f$ is a continuous function (not uniformly continuous) on $(0,1)$ then it can't be approximated ...
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Let $f$ is a uniformly continuous function on $(0,1).$ Is it possible to approximate $f$ by polynomials.

Let $f$ be a uniformly continuous function on $(0,1)$. Then $f$ can be extended to a continuous function $\widetilde{f}$ on $[0,1]$. By "Weierstrass Approximation Theorem" the extended function ...
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$\exp_{x}$ is only $C^{1}$ at $y=0$.

According to the following image of book "Riemann-Finsler geometry" by chern & shen I would like to know which theorem of ODE theory is applied? Thanks.
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Simple question about discrete metric and openness.

You may think this is silly question, but I'm really confused. In discrete metric, every singleton is an open set. And, the proof goes like this $\forall x \in X$, by choosing $\epsilon < 1$, ...
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Show that any two norms on a finite dimensional vector space $V$ over the set of real numbers are equivalent.

I know that the question has already an answer. But, I am trying to do it in a different way:- I am using the fact that any two norms on $\mathbb{R}^n$ are equivalent. Let us assume that the $dim ...
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A doubt in the theorem of equivalence of two normed spaces.

Theorem:Let $(X,\|\|_1)$ and $(X,\|\|_2)$ are normed spaces. Let $\|\|_1$ and $\|\|_2$ are equivalent $<=>$ there exist $c_1,c_2>0$ such that $c_1\|x\|_1\leq\|x\|_2\leq c_2\|x\|_1$ , $\forall ...
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How to prove that the given set is not uncountable?

I was trying to solve the question given in my assignment on metric spaces. Let $S$ be a subset of $R$. Let $C$ be the set of points $x$ in $R$ with the property that $S\cap (x-\delta,x+\delta )$ is ...
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Does this proof of $x\in E'\rightarrow \exists \{x_n\}\subseteq E : x_n\to x$ use the axiom of choice?

Let $(X,d)$ be a metric space. Let $E$ be a subset of $X$. If $x$ is a limit point of $E$, then there exists a sequence $x_n\in E$, $n= 1,2,\dots$ such that $x_n\to x$. Proof. Pick ...
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Metric space extension

Is single point extension of a metric space possible? Let $(X,d)$ be a metric space and $\overline{X}=X\cup \{\overline{x}\}$. Is it possible to find a metric $\overline{d}$ for which ...
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$f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ continuous ?

Let $f:\mathbb R \to \mathbb R^n$ be a function whose graph $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , then is $f$ continuous ?
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Metric on infinite cartesian product $\mathbb{R}^w$ and convergent series

When considering a metric for the infinite cartesian product $\mathbb{R}^w$, in Munkres's Topology (2nd edition, p.124) mentions that: Why is it that $d(x,y)$ "does not always makes sense, for the ...
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Let $\{K_i\}_{i=1}^{\infty}$ a decreasing sequence of compact and non-empty sets on $\mathbb{R}^n.$ Then $\cap_{i = 1}^{\infty} K_i \neq \emptyset.$

Let $\{K_i\}_{i=1}^{\infty}$ a decreasing sequence of compact and non-empty sets on $\mathbb{R}^n.$ Then $\cap_{i = 1}^{\infty} K_i \neq \emptyset.$ I heard about a proof that take $x_i \in K_i.$ ...
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Example of metric space with given property

Give an example of metric space that contains balls $B(x_1,r_1)\subsetneqq B(x_2,r_2)$, with $r_1>r_2$. Was initially thinking about discrete metric, however, in discrete case one can never ...
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If $C$ is the Cantor set, then $C = \text{bd}(C)$?

Let $C$ bethe Cantor set, then is it true that $C = \text{bd}(C)$? I know that the $C$ is closed since it the intersection of closed intervals, which is always closed. This means that $C$ contains ...
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Does the metric in the Theory of Relativity actually satisfy the definition of a metric?

Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics ...
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Why study dimensions?

I am quite new to the forum so please feel free to correct/give pointers if I am posting something in the wrong place. I have been perusing "Dimension Theory" by Witold Hurewicz & Henry Wallman ...
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Prove that $A^c$ closed $\Rightarrow$ for all $a\in A$ there exists $r>0$ such that $B(a,r)$ is contained in A.

Let $(X,d)$ be a metric space and $A$ is a subset of $X$. $A^c$ is complement of $A$ in $X$. Use only the following characterization of closed sets: $$A \text { is closed if it contains all it's ...
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Proof of theorem $20.5$ Munkres Topology

First the metric on $\mathbb R^{\omega}$ is defined as $$D(x,y)=sup\left\{ {\bar d(x_i,y_i)}\over i \right\}$$ where $\bar d(x,y)=\min\{d(x,y),1\}$ and $\bar d$ is the Euclidean metric on $\mathbb ...
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Regarding the Manhattan metric and the projections of a closed ball.

In $R^2$ If I have the closed ball $\bar{B}(0,1)$ in the Manhattan metric and I take the set $A_x = \{ y \in R| (x,y) \in \bar{B}(0,1) \} $ and $P_A = \{ x \in R | (x,y) \in \bar{B}(0,1) \}$ (the ...
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Showing that If $A$ and $B$ are closed disjoint subsets of a metric space then there exists disjoint open sets containing $A$ and $B$.

Suppose $A,B$ are closed subsets of a metric space $(X,d)$. I am trying to show that there exist disjoint open sets $U,V$ such that $U$ contains $A$ and $V$ contains $B$. I managed to find an open ...
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“Accumulation” points of a convergent net

Let $(X,d)$ be a metric space, $D$ a directed set and $\phi :D \rightarrow X$ a net converging to some $x_0 \in X$. Can there be an increasing sequence $\{d_n\} \subset D$ s.t. $\{\phi(d_n)\}$ ...
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Show that a sequence converges in on metric iff the sequence converges in another metric.

Let $\delta(x,y) = \bigg | \frac{1}{x}-\frac{1}{y} \bigg|$ and $d$ is the usual euclidean metric. Show that $(x_n)$ converges to $a$ using $\delta$ iff it converges to $a$ using $d$. My attempt: ...
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Metric space $M = M_1\times \cdots M_n$ with metric $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$. Showing triangular inequality.

I have the following metric space: $M = M_1\times \cdots \times M_n$ with metric $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$ where $d_i$ is the metric for each $M_i$, and ...
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Pick $b\notin B[a,r]$ show that there exists $s>0$ such that $B[a,r]\cap B[b,s]$ is empty

I need to solve this question: Pick $b\notin B[a,r]$ show that there exists $s>0$ such that $B[a,r]\cap B[b,s]$ is empty My idea is to suppose a point $p$ in the intersection. Then ...
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Polar coordinates in taxicab geometry

We know that in euclidean $\mathbb{R}^2$ space polar coordinates are defined by $$r = \sqrt{x^2 + y^2}$$ $$\theta = \arctan\frac{y}{x}\text{.}$$ Now, geometrically we can think of it as of point, ...
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$\Delta = \{(x,x), x\in M\}\subset M\times M$. Show that if $z\in M\times M - \Delta$ then there is a ball with center $z$, disjoint from $\Delta$

I need to show this: $\Delta = \{(x,x), x\in M\}\subset M\times M$. Show that if $z\in M\times M - \Delta$ then there is a ball with center $z$, disjoint from $\Delta$ I need to use the metric ...
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metric spaces - basic inequality

Let $(\Omega, d)$ be a metric space. I have to show that $ d(\alpha ,\beta) \ge | d(\alpha, \theta) - d(\theta, \beta)|$ for every $\alpha, \beta, \theta \in \Omega.$ Starting with the triangle ...
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If G is an open subset of a metric space, is it true that $\text{int}(\bar{G}) = G$?

If G is an open subset of a metric space, is it true that $\text{int}(\bar{G}) = G$? I have not been able to find a counter example or a proof. Any hints?
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Looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism

I am looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism . Please help . Thanks in advance .
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How to proof an open set and its complement contained in a metric space with infinite element are both infinite? [duplicate]

Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement $U^c$ = X\U are both infinite.
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How does the “arc tangent metric” $d(x,y) = | \arctan(x) - \arctan(y)| $ work?

I see there are some counterexamples and so forth in metric spaces regarding the metric $$d(x,y) = | \arctan(x) - \arctan(y)| $$ But honestly I have no intuition as to how it works For example, in ...
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Is it always true that ${c(\bar A)} = \overline{c({A})}$?

Suppose $(X,d)$ is a metric space. If $A$ is a closed subset of X then ${c(\bar A)} = \overline{c({A})}$ where $c$ is the complement of the set and $A$ is a subset of the metric space. I think ...
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Let $f: X \subset \mathbb{R}^n\to \mathbb{R}^m$. Then, $f$ is uniformly continuous if, and only if, for every sequence…

Let $f: X \subset \mathbb{R}^n\to $. Then, $f$ is uniformly continuous if, and only if, for every sequence, $(x_n)_{n\in\mathbb{N}}$, $(y_n)_{n\in\mathbb{N}}$ such that $d(x_n,y_n) \to 0$, then ...
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Open ball cartesian product on metric space: $B(a,r) = B(a_1, r)\times \cdots \times\ B(a_n,r)$

I need to prove that $$B(a,r) = B(a_1, r)\times \cdots \times B(a_n,r)$$ in $M=M_1\times\cdots \times M_n$ where $M_i$ is a metric space and the metric is $d''(z,z') = \max\{d_i(x_i,y_i), i \in ...
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Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$? [duplicate]

Is the set $\{\big(x,\sin(1/x)\big):x\ne 0 \}$ connected in usual metric of $\mathbb R^2$ ? I tried writing it as a union of two connected sets , or otherwise as a union of two disjoint non-empty ...
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Interior and Exterior Points in C[0,1] with Supremum Metric

Let $C[0,1]$ be the set of real, continuous functions on the set $[0,1]$ with the metric \begin{equation} d(f,g)=sup_{x\in [0,1]} |f(x) - g(x)|. \end{equation} Consider the set $C$ of constant ...
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Sequences in metric spaces.

Given , $X= l_p (p\geq 1)$ , and let $d(x,y) = ( \sum_{k=1}^{\infty} |x_k - y_k |^{p})^{\frac{1}{p}}$ where $x= \{x_k\}_{k\geq 1}$ and $y= \{y_k\}_{k\geq 1}$ are in $l_p$. Let $\{x^{(n)}\}_{n \geq ...
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Metric spaces whose open sets form a $\sigma$-algebra.

I have the following question Characterise the metric spaces whose open sets form a $\sigma$-algebra. I apologise if this seems like too basic of a question to ask here, but seeing as English is ...
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Find $\text{dist}(i,A)$ where $A = \{z\in\mathbb{C}:|z-(1-i)|<1\}$

I am trying to find $\text{dist}(i,A)$ where $A = \{z\in\mathbb{C}:d(z,(1-i))<1\}$ where $d$ is the usual metric on $\mathbb{C}$, that is d(x,y) = |y-x|. I know that the set A in the complex plane ...
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Completion of polynomial space with max norm

Let's start with space of finite polynomials over $\mathbb{R}$. Weierstrass theorem says it's dense in $C[0;1]$ by norm $\|P(t)\|=\max|P(t)|, t \in [0,1]$. So the completion of this space is $C[0,1]$. ...
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Proving Corollary to Riesz's Lemma

Let $(X,\|\cdot\|)$ be a normed linear space and $Y \leqslant X$ be a proper subspace. If $\text{dim}(Y) < \infty$, show that there exists $x \in X$, with $\|x\| = 1$ such that $d(x,Y) = 1$. ...
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Example for converges series in the metric space

Give example for converges series in the metric space: $$ \quad\quad\quad(\mathrm {R}^n,d_{\infty}),d_\infty=\max\mid x_i-y_i\mid$$ My attempt: Let ...
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Distance inequality

Consider $H=\{ (x,y) \in {\bf R}^2\mid x$ or $y$ is an integer $\}$ If $d$ is canonical distance in ${\bf R}^2$, show that if $d(x):=d(x,H)$, (1) $$ d(x) - d(y) \leq d(x,y) $$ if $x,\ y$ are in same ...
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How do I show that the open rectangle is convex?

I have no ideia how to solve this. How to show that the open ball is convex or the open cube is, that`s easy, but how to show that the open rectangle is? (The same holds for the closed rectangle). ...
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Compact subsets of metric space with French railway metric

Let $A=\{0,1,2,...\}$ with $f$ the French railway metric that has centre $0$ and $f(a,0)=1$ for all $a\in A$ with $a\neq0$. How do I show that the metric space $(A,d)$ is complete? How do I ...
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Are $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$ or $d(f,g)=\min \limits_{a\leq x \leq b} |f(x)-g(x)|$ metric?

Neither $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$ nor $d(f,g)=\min \limits_{a\leq x \leq b} |f(x)-g(x)|$ meet the triangle equality condition to be metric. Because if you some $x$ (say $x_1$) ...
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Is distance function defined on a convex set is always convex?

I am looking for an answer to the following question: Is the distance function defined on a convex set always convex? Obviously the convex set in question is metric. In particular I am interested ...