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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I've got a definition, but says something strange, what does it mean?

$M_1:=(M,d_1), \ \ M_2:=(M,d_2)$. $d_1$ is equivalent to $d_2$ if the identity $x\rightarrow x$ of $M_1$ over $M_2$ is an homeomorphism I'm not sure what it is talkin about when it says "identity" ...
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Help understanding a proof that the metric space of bounded functions is complete.

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...
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Why does the second statement follow from the first one?

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...
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Showing that the given interval is is connected.

Let $(R,d)$ be the space of real numbers with the usual metric. Let $I$ be an interval such that $I \subseteq R$. We need to show that $I$ is connected. Let us say that $I$ is not connected , thus ...
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Does every connected metric space , with more than one point , contains a path connected subset with more than one point ?

Does every connected metric space , with more than one point , contains a path connected subset with more than one point ? Is there any additional condition imposing which on the mother space will ...
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23 views

Let $I = [0, 1) ∪ [2, 3]$ with the usual metric.Prove that the closure of $B$ is not the closed ball of center $0$, radius $2$.

Let $I = [0, 1) ∪ [2, 3]$ with the usual metric. Let $B = I(0, 2)$ be the open ball of center $x$ and radius $1$. Prove that the closure of $B$ is not the closed ball of center $0$, radius $2$. The ...
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Difference between a continuous function and an isometry? Is a continuous function a homomorphism?

Definition of continuous function on a set: Suppose $X$ and $Y$ are metric spaces. Let $f: X \to Y$. We say $f$ is continuous on $X$ if for every $\varepsilon >0$, $\exists \delta >0$ such that ...
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$k$-Lipshitz function $f: (E,d_E) \to (F,d_F)$ prove if $f$ is k-Lipshitz then $f$ is continuous

$k$-Lipshitz function $f: (E,d_E) \to (F,d_F)$ prove if $f$ is k-Lipshitz then $f$ is continuous where $ (E,d_E), (F,d_F)$ are metric spaces $f$ is said to be $k$-Lipshitz if for $f : E \to F$ for $k ...
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Let $A \subset X$ metric space. Then $d(x,A) = 0$ if and only if $x\in \overline{A}?$ [duplicate]

I am trying to prove this. I did not find in any book. I was making some exercises where I had to prove that if $A$ is closed and $x\not\in A$ then $d(x,A) >0.$ Because of the condition "being ...
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Interior of cartesian product is cartesian product of interiors

I have to prove that: $$Int(A\times B) = Int(A)\times Int(B)$$ Where $A\subset M$ and $B\subset N$, both $M$ and $N$ metric spaces. The problem is that the exercise does not specify the metric, so ...
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2answers
31 views

How to show a topological space is metrizable given a metric

My question is rather general and I know there are similar questions but none of them seem to give a detailed answer. My question is suppose we have a topological space $(X,\mathcal{J})$ and we are ...
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28 views

Uniformly continuous function induces another uniformly continuous function

Let $(X,d)$ be a metric space and $f: X \to X$ is uniformly continous. Define $\varphi: X \to \mathbb{R}$ such that $\varphi(x) = d[x,f(x)],\forall x \in X$. Does $\varphi$ need to be uniformly ...
2
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3answers
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Why the interior of $\mathbb{Q}$ in $\mathbb{R}$ is empty?

I don't understand why the interior of $\mathbb{Q}$ in $\mathbb{R}$ is empty, since, for every ball with the center being a rational number, given an $\epsilon>0$, I can find an infinite sequence ...
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Is there a connection between limit point of a subset of a metric space and the limit of a function?

Is there a connection between limit point of a subset of a metric space and the limit of a function, or limit of a sequence? I am not sure but I don't think there is because there can be more than ...
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Is it true that if $f$ is a continuous function on $(0, 1)$ can be approximated by polynomials(Weierstrass theorem)?

I have found a counter example. Consider the function $f(x)=1/x$ on $(0,1)$. It is unbounded. So, $f$ can't be approximated by polynomials because polynomials are bounded on a bounded interval. But ...
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Can anyone tell me what is the difference between strong and weak contraction (metric spaces)?

In my assignment a question was given on strong contraction. I could understand the meaning and searched in wikipedia and also in books but I could find the definition of it. Can anyone give the ...
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How to confirm that if a function $f\circ f$ is a strong contraction, then $f$ has a fixed point or not?

Suppose that $(X,d)$ is a complete metric space and $f:X \rightarrow X$ is such that $f\circ f$ is a strong contraction. Must $f$ have a fixed point? So, it is given that $f\circ f$ is a strong ...
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Prove expansive function on a compact set is surjective.

Let $M$ be a compact set and $(M,d)$ be a metric space, define function $f:M\to M$ such that for all $\,p,q\in M$ $$d(f(p),f(q))\ge d(p,q)$$ Prove $f$ is surjective. I observed that compactness ...
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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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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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59 views

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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1answer
21 views

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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35 views

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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108 views

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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1answer
46 views

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 ...