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Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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Closed spaces in a metric space.

Let $(X,d)$ a metric space. Let $F$ and $A$ two subsets of $X$ such that $A\cap F=\emptyset$ and $F$ is closed. Suppose that for any converging sequence $\{u_{n}\}_{n\in \mathbb{N}}\subset A$, we have ...
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Continuity preserves limits in metric spaces [on hold]

Let M be a metric space, $\mathit{f}: M \rightarrow \mathbb{R} $ continuous and a convergent succession $x_{n}$ in M, proof that: $$ \lim_{n \to \infty} f(x_{n}) = f(\lim_{n \to \infty}x_{n}) $$
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Is this exotic function actually a metric?

I've got this pretty exotic metric of which I cannot seem to prove the triangle inequality. Given that I already have a metric $\delta$ on the unit ball in $\mathbb{R}^n$, I define a new metric $d(x,y)...
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If $A_q$ is open for all $q \in \mathbb Q$ prove $A_r$ is open for all $r \in \mathbb R$

The problem is as follows. Let $X$ be a metric space and $f: X \to \mathbb R$ be a function such that $A_q = \{x \in X: f(x) < q\}$ is open for all $q \in \mathbb Q$. Prove $A_r = \{x \in X: f(x) &...
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Are Two Metric Spaces Equivalent?

Are the following metrics equivalent on $\Bbb R$? $ d(x,y)=|x-y|$ and $d'(x, y)=|\tan^{-1}(x) - \tan^{-1}(y)|$.
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Compact + chain-connected metric space => connected

How to show that if $M$ (a metric space) is compact and chain-connected then it is connected ? Definition of $\varepsilon$-chainable : $(X,d)$ is $\varepsilon$-chainable if given any two points $a,b\...
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What set $Y$ that $C_{c}(U) \subset Y$ and $Y$ separable w.r.t. $\vert \vert \cdot \vert \vert_{\infty}$

Let $U \in \mathbb R^{d}$ open and further $C_{c}(U)$ be the space of functions with compact support. Show that $C_{c}(U)$ is separable w.r.t. $\vert\vert \cdot \vert \vert_{\infty}$, and I have been ...
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Completeness of $\ell^2$ space

I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a ...
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1answer
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Is $\overline{B_1((1,0))}\cup B_1((-1,0))$ connected? Is it path connected?

I'm trying to solve a question which asks me to let $B_1(p)$ denote the unit ball around $p$ in $\mathbb{R}^2$. I'm supposed to decide whether $\overline{B_1((1,0))}\cup B_1((-1,0))$ is connected and/...
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The normed vector space of continuous function is complete

First of all, let $(f_n)$ be a Cauchy sequence in $B(x)$ which is the vector space of bounded functions $f\colon X \to \mathbb R$ equipped with the norm $\|f\| = \sup|f(x)|$. Note that $|f_n(x)-f_m(x)...
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Prove the following set is open in $\mathbb R$ with the standard metric.

Let $x_k = (\sin k, \arctan k, k^3)$, $k \in \mathbb Z_+$. Prove (by the open ball definition) that $V = \mathbb R^3\setminus\{x_k: k \in \mathbb Z_+\}$ is open in $\mathbb R$ with the standard metric....
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A metric space in which $3^\infty=2^\infty=0$

I want a space containing all the positive integers in which $3^nx+3^n-2^n\to0$ as $n\to\infty$ Perhaps paradoxically, numbers not factorisable by $2,3$ would be a sufficient set for me (in case that ...
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Proving that a certain space of compact sets is a complete metric space

Let $(X,d)$ be a metric space, and let $K(X)$ denote the collection of all non-empty compact subsets of $X$. Define a function, $d_h\colon K(x)\times K(x)\to\mathbb R$ by letting $$d_h(A,B)=\inf\{\...
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Existence of common convergent subsequence indexes for infinite number of sequences in a compact metric space.

A slight infinite extension of this Show that two bounded sequences have convergent subsequences with the same index sequence Let $S$ be a compact metric space. Suppose, for each $m\in\mathbb{N}$, $\...
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1answer
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If $U$ and $V$, having $U \cap V = \emptyset$, are open in $\mathbb R$ with the standard metric, then there exists $c \notin U \cup V$

If $U$ and $V$, having $U \cap V = \emptyset$, are open in $\mathbb R$ with the standard metric, then there exists $c \notin U \cup V$ Proof: Let $A = [a,b] \cap U$ where $a \in U$ and $b \in V$. ...
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Natural Extensions of the $p$-Adic Norm to Higher Dimensions

There are two classes of completions for $\Bbb{Q}$, we get $\Bbb{R}$ by considering Cauchy sequences with respect to the standard Euclidean metric, and we get the $p$-adic numbers, $\Bbb{Q}_p$ when ...
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1answer
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Metric spaces bounded sets equivalence

Let X be a a non empty metric space and suppose A $\subseteq$ X. Then A is bounded if $\exists$ $z$ $\exists$ $r>0$ so that $A \subseteq B(z,r)$. WTS The following are equivalent: 1) A is ...
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Example of a strong b-metric which is not a metric.

Definition: Let $X$ be an arbitrary set, $d:X\times X\to [0,\infty)$ be a mapping satisfying: (a) $\forall_{x,y\in X}\; d(x,y)=0\iff x=y$; (b) $\forall_{x,y\in X}\; d(x,y)=d(y,x)$; (c) $\exists_{s\...
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surface area of a solid of revolution. (request for a source for rigourse proof )

link: Doubt in Application of Integration - Calculation of volumes and surface areas of solids of revolution. I know that this question has been asked and answered before , but non of the answers ...
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If $K$ and $N$ are metric spaces with $K$ compact and $N$ complete, then $\mathcal{C}(K,N)$ is a complete metric space

Let be $K$ a compact metric space, $N$ a complete metric space and $\mathcal{C}(K,N)$ the space of continuous function on $K$ to $N$. I'm trying to prove that $\mathcal{C}(K,N)$ is complete with ...
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Questioning dense subset completeness (counterexample)

Let $X$ be a separable metric space and $A \subset$ X be countable and dense. Characterize the statements below as true or false (and why). If every Cauchy sequence in $A$ converges in $X$, $...
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$\lim_{p\to\infty}{\ell^p}$ and how this relates to the maximum metric.

If the $\ell^p$ norm metric is defined as $$ \ell^p=\left[\sum_{n=1}^{m}{x_{n}^p}\right]^{1/p} $$ and the $\ell^\infty$ norm metric is defined as $$ \ell^\infty=\max(|x_1|,|x_2|,|x_3|...,|x_{m-1}|,|...
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Boundary of a set in metric space

I'm trying to find the boundary of a set in order to check whether or not the set is open or closed on the metric I'm given. While I understand the latter, I'm not sure how to properly understand the ...
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1answer
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recurrent sequence in a compact space

please how to solve this question Let $E$ a compact metric space and a function $f:E\to E$ such that $$ \forall x,y \in E, d(f(x),f(y))\geq d(x,y)$$ Let $a\in E$ and a sequence $(f^n(a))$ , ...
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Complete space of functions with $f(a)=f(b)$?

Consider M is space of continuous functions (on $[a,b]$) with condition: $f(a)=f(b)$ . Is it complete metric space with $\mu(f,g) =\max\underset{x\in [a,b]}{|(f(x)-g(x))|} $? In my opinion it's true. ...
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Homeomorphism between $([0,1]^2, \delta )$ and $[0,1]^2$ with standard euclidean metric

I am stuck as to how I find the homeomorphism described above. $\delta$ is here described to be the metric $$\delta((m_1,n_1), (m_2,n_2))= max\{d_M(m_1,m_2),d_N(n_1,n_2)\}$$ a metric on $M\times N$ ...
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1answer
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Convergence of measure on clopen basis implies the existence of a limit measure?

Let $X$ be a compact metrizable space. Let $B$ be basis for the topology of $X$. Assume that all sets in $B$ are clopen. Let $(\mu_n)_{n=1}^{\infty}$ be sequence of Borel regular probability measures ...
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How do I visualize a set in metric space?

If I'm given a metric, say the discrete metric $\text d_0(x,y):=\begin{cases} 0, & \text{if }\vec x= \vec y\\ 1, & \text{if } \vec x \neq \vec y\ \end{cases}$ and want to visualize a set ...
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A sequence of continuously differentiable functions with non-differentiable limit

Working in the metric space $C[a,b]$, the set of continuous functions $f:[a,b]\rightarrow[a,b]$, with the supremum metric, I need to demonstrate that $C^1[a,b]$, the subset of continuously ...
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Separability of a product metric space

I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is ...
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Equality of certain distances in a normed vector subspace

Let $E$ be a normed vector space and $F$ a vector subspace of $E$. If $y \in F$, $x \in E$ and $0 < a \in \mathbb R$, prove that $d(y + ax, F) = ad(x,F)$. I've tried to write down the definitions ...
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Metric on Lorentzian manifold

Since the signature of a Lorentzian manifold (M,g) is (-,+,+,+), am I right to assume that the determinant of $g_{ij}$ with EACH coordinate system is $-1$?
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A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. The set $\Omega$ is contained in some large disc.

My complex analysis textbook says the following: A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. In other words, the set $\Omega$ is contained ...
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Proof of Schur's Theorem

On Pg. 123 of Schaum's Tensor Calculus: At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$ for any ...
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1answer
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Obtaining Density From Homeomorphism onto Image

Let $f:X\rightarrow Y$ be a continuous function between metric spaces which is a homeomorphism onto its image and let $K\subseteq X$ be non-empty and $D\subseteq Y$ be dense and satisfy $$ D\cap {f(X)}...
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Thin Metric Space

Let $X$ be a complete thin metric space and let $A, B$ be disjoint closed connected subset of $X$ then there is a compact set $K$ such that each neighborhood of $K$ disjoint from $A \cup B$ separates $...
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Metric for the shortest path between two cells of a class I octahedral Goldberg polyhedron.

Consider the following class I octahedral Goldberg polyhedron, $GP_{IV}(9,0)$: Every cell of this polyhedron can be uniquely determined using three coordinates, $x, y, z \in \mathbb{Z}$, such that $\...
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Show that there is $x\in[-1;1]$ such as the set of limit point of $u_n(x)$ is $[-1;1]$ where $u_0(x) = x, u_{n+1}(x) = 2u_n(x)^2 - 1$

Let be $f$ defined by $f(x) = 2x^2 -1$. Let be $(u_n(x))$ the sequence defined by: $u_0(x) = x, u_{n+1}(x) = f(u_n(x))$ Show that there is $x\in[-1;1]$ such as the set of limit point of $u_n(x)$ is ...
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Relation between total variation and KS distance between measures on $[0,1]^d$

Let $P$ and $Q$ be two probability measures on the space $[0,1]^d$, $d \in \{1, 2, \ldots \}$, endowed with the $L_\infty$ norm and the corresponding Borel $\sigma$-field, $\mathcal{B}$. Let $$F_P(\...
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1answer
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Define a metric on collection of compacts

Let $(\mathbb{R}^2,d)$ be a metric space, where $d$ is the Euclidean metric on $\mathbb{R}^2$. Let $\kappa$ denote the set of compact subsets of $R^2$. Which one of the following expression defines a ...
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Prove that $f:X \rightarrow \mathbb{R}$ is constant. [duplicate]

Let $(X,d)$ be a connected metric space and $f:X \rightarrow \mathbb{R}$ be a continuous function. Suppose that for every $x \in X$ there exists an open set $U \subseteq X$ so that $x \in U$ and ${f|}...
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Metric topology induced by the sum of two metrics

I have to show the following: Let $X$ be a set with metrics $d_1$ and $d_2$ inducing metric topologies $\tau_1$ and $\tau_2$. Define a new metric on $X$ where $d(x,y) = d_1(x,y) + d_2(x,y)$ for ...
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How is the metric topology the coarsest to make the metric function continuous?

Let $X$ be a metric space with metric $d$. If $\mathcal{T}$ is a topology on $X$ such that the function $d\colon X \times X \to \mathbb{R}$ is continuous, then how to show that $\mathcal{T}$ is finer ...
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Convergence of a sequence in metric space

Can someone please help me with this problem? Thanks! Check if the sequence $x_n= (1+1/n)^n$ is convergent in $ (X,d)$ where $d(x,y)=$ $\frac {2|x-y|}{3+2|x-y|}$, and if it is convergent, then find ...
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1answer
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Metric space which is totally bounded is separable. Baby Rudin Ex 2.24

Let $X$ be a metric space which is totally bounded. Show that $X$ is separable. A metric space is called separable if it contains a countable dense subset. A metric space is called totally bounded ...
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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 (source: picofile.com) I would like to know which theorem of ODE theory is applied? Thanks.
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How to prove $C_{c}^{\infty}[a,b]$ a complete metric space [duplicate]

How do you prove the space of smooth functions with compact support in an interval $[a,b] \subset \mathbb{R}$, with the metric $\rho(\varphi_{1},\varphi_{2})= \sum_{n=0}^{\infty}2^{-n}\frac{\left \| \...
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Do these define metrics on $\mathbb{R}$?

Which of the following define a metric on $\mathbb{R}$? $$d_1(x,y) = \frac{\bigl||x|-|y|\bigr|} {1+|x||y|},$$ $$d_2(x,y) = \frac{\bigl||x|+|y|\bigr|} {1+|x||y|}.$$ I think that both option 1 and ...
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1answer
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Is $X$ totally bounded?

I came across the following claim in this post. Let $X$ be a metric space in which every infinite subset has a limit point. Then for every $\delta>0$ there exists $N_{\delta}\in \mathbb{N}$ and $...