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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Looking for an incomplete metric space $X$ , all whose singletons are open

Does there exist an incomplete metric space whose metric topology is discrete ( not the discrete metric ) ? Please help . Thanks in advance
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Dense in X,and approaches to confirm denseness inside a set. [closed]

If a subset P of X is everywhere dense in X then P complement in X has empty interior. True/False.(Please with reasons)
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Does $d(x,y)=\sqrt[3]{|x_1-y_1|+|x_2-y_2|}$ define a metric?

$d: \mathbb{R^2} \times \mathbb{R^2} \rightarrow \mathbb{R}$ where $d(x,y)=\sqrt[3]{|x_1-y_1|+|x_2-y_2|}$ for $x=(x_1, x_2)$ and $y=(y_1, y_2)$ I am trying to determine if $d$ defines a metric on ...
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Completion of two metric spaces

Let $X=\mathbb{R}\setminus\{0\}$ be equipped with two metrics, namely $d_1(x,y) = |x-y|$ and $d_2(x,y)=|\frac{1}{x}-\frac{1}{y}|$. I am looking for the completion of these two spaces. My attempt: ...
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Triangle Inequality of the Cartesian product with Max function

Let $(X,d)$ be a metric space. Define $$d'((x,y),(z,w))=max\{d(x,z),d(y,w)\}.$$ I'm trying to prove the triangle inequality for this, but really don't have a clue how. Any tips or suggestions ...
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Does it make sense to define a “metric topological space” $(M, d, \tau)$

When doing things related to compactness, sometimes you have to switch definition from sequential compactness which is defined on a metric space $(M, d)$, to things related to covering compactness ...
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Conceptual Question on Takens embedding Theorum

I am from signal processing background and so unaware of many details of Takens phase space reconstruction theorum. Reading the paper : A First Analysis of the Stability of Takens’ Embedding download ...
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Show that the topology $T_d$ induced by metric $d$ is not coarser than the topology $T_e$ induced by metric $e$.

Problem: Let $C[0,1]$ denote the collection of all real continuous functions defined on $I=[0,1]$. Consider the metrics $d$ and $e$ on $C[0,1]$ defined by $$d(f,g) = \sup\{|f(x) - ...
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Separability inside seprable set [duplicate]

If the metric space $X$  is separable, then there exists a countable dense subset $P$, but if $X$ is uncountable, prove that $X-{P}$ is not separable.
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Find distance to a set (subspace) without computing closest point

General setup: we have a finite-dimensional normed linear space $(V, \| \cdot \|)$, a subspace $U \subset V$, and a fixed vector $v_0 \in V$. We want to find the distance between $v_0$ and $U$. (No ...
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Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
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Uniformly continuous independent of metrics?

Let $(X,d)$ and $(Y,e)$ be metric spaces. A map $f:X\to Y$ is uniformly continuous if for each $\epsilon>0$ there exists $\delta >0$ such that whenever $d(x,y)<\delta$ we have ...
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Proof of the Arzelà–Ascoli Theorem

I'm stuck on a particular line of the proof of The Arzelà–Ascoli Theorem. In lectures, we have: $1.$ Defined equicontinuous as: Let $X$ be a metric space, $C(X) = \{f: X \rightarrow ...
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Proof of the second property of metrics for metric space $(C^{\infty}[a,b],\rho)$

Let $C^{\infty}[a,b] $ the set of all infinitely differentiable functions on $ [a, b] $, and let for $x,y \in C^{\infty}[a,b]$, $$ \rho(x,y)=\sum_{k=0}^{\infty}\frac{1}{2^k}\cdot ...
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Give example of congregate serieses in the metric space : $(R^n,d_1),d_1=\sum_{i=1}^{n}\mid x_i-y_i \mid$

Give example of congregate serieses in the metric space : $$(R^n,d_1),d_1=\sum_{i=1}^{n}\mid x_i-y_i \mid$$ What I tried: I think I should find $\{X_n\}\to x$ $\left(\frac{\sin ...
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Fixed Point with two equivalent metric complete spaces

Can anyone help me with this: Let $(X,d)$ and $(Y,d')$ be two metric spaces. We know that if the metrics $d$ and $d'$ are equivalent (strongly, that is), the completeness of one implies that of the ...
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36 views

Contraction mapping theorem for metric spaces - why is this part of the proof notable?

Someone has made a remark to me that from the proof of the CMT for metric spaces we get that $$d(x, x_n) \leq \frac{K}{1-K}d(x_{n-1}, x_n).$$ What is this saying? Why is this remark worthy? (The ...
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What is “the set in which this* satisfies lipschitz condition”?

I'm very confused, I believe my teacher was not clear enough, maybe im missing something. $$1)y''+2ay'+by=0$$ In my "definition of lipschitz condition": G is the domain of a function f in which you ...
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Prove $(\Bbb R^{n},d_{p})$ forms a complete metric space

Prove $(\Bbb R^{n},d_{p})$ forms a complete metric space So I know that in order for a metric space to be complete I must show that every Cauchy sequence in $\Bbb R^{n}$ converges on $\Bbb R^{n}$ ...
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Is the set of polynomial dense in $C[-1, 0]$?

If the set is defined as $$\{a_0+a_1x+a_2x^2+...+a_nx^n, \text{where }n\ge 0 \text{ and } a_0+a_1+...+a_n=0\},$$ is the set dense in $C[0,1]$ and $C[-1,0]$? For the first question, I'm thinking ...
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Isometries of metric spaces $Z=X\cup (X\times\mathbb{R})$ (corrected)

Let $(X,d_1)$ be a compact metric space and consider $Z=X\cup (X\times\mathbb{R})$. Consider the metric $d$ on $Z$ defined by: $$d(x_1,x_2)=d_1(x_1,x_2)$$ $$d(x_1,(x_2,t_2))=d_1(x_1,x_2)+1$$ ...
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Arzela-Ascoli Theorem on metric spaces

I've been looking for a proof of one particular direction of this theorem for metric spaces. I've looked online, but everyone seems to use different terminology/notation to state the theorem, so I'd ...
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open ball on metric $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$ in $\mathbb{R}^2$

I have the following metric: $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$ in $\mathbb{R}^2$ What figures form the open and closed ball? What about the sphere? That's what I thought: ...
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If $X$ is a countable dense subset of the separable metric space $P$, then $P \setminus X$ is not separable [closed]

If the metric space $P$ is separable, then there exists a countable dense subset $X\subset P$.but if P is uncountable, Prove that $P \setminus X$ is not separable.
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$X,Y$ metric spaces , $X$ complete , $Z$ is Hausdorff , $f,g:X \times Y \to Z$ continuous in each variable and coincide on a dense subset , is $f=g$?

Let $X,Y$ be metric spaces , $X$ be complete and $Z$ be a Hausdorff space ; let $f,g:X \times Y \to Z$ be functions such that each of $f$ and $g$ is continuous in $x \in X $ for each fixed $y \in Y$ ...
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A notion of nonpositive curvature for general metric spaces

The proof of the following result should be done by using the second variation formula of geodesics but I do not know how to start or what is the main idea of the proof. (Lemma 3.7 in the paper: A ...
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Prove that $\mathbb{R}$ is not isometric to any proper subset of $\mathbb{R}$.

The question is in the title. I can show easily that $\mathbb{R}$ is not isometric to any closed interval $[a,b]$ in $\mathbb{R}$, but I am struggling to show it for any arbitrary subset.
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Completeness and Compactness of Cartesian Product of Metric Spaces

Suppose $\{(X_i,d_i)\}_{i\in \mathbb{N}}$ is a collection of metric spaces with the cartesian product defined by $A=\prod_{i=1}^{\infty}X_i$. Let the metric on $A$ be given by ...
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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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34 views

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 ...