Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Why is $[-1,1]$ compact when $a_n = (-1)^k$ does not converge in $A$

I know this question sounds silly but I was reading the definition of compactness and couldn't quite wrap my head around this Compactness :A subset $A$ of a metric space $M$ is compact if every ...
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Is $T$ a homeomorphism?

Let $X$ be the space of all polynomials in one variable over $\Bbb R$. If $p=a_0+a_1x +a_2 x^2+...+a_n x^n$,define $||p||=|a_0|+|a_1|+...+|a_n|$. Which are correct? $(X,d)$ is complete where ...
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Why is the function $f(x)=x^2$ is not a contraction on $[0,0.5]$?

Let $(X,d)$ be a metric space and let $F:A(\subset X)\to X$. We say $F$ is a contraction if there exists $\lambda$ where $0\leq\lambda<1$ such that $$d(F(x),F(y))\leq\lambda d(x,y)$$ for all ...
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How can one measure distance between point and the line in maximum metric space?

Given metric space $M = (\mathbb{R}^2, d)$ where $d = \operatorname{max}\{|x_1 - y_1|, |x_2 - y_2|\}$, how can one measure distance from some arbitrary point $X$ to the line $y = 3$, let's say? How ...
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If $X \setminus A$ is disconnected then prove or disprove $X \setminus B$ is also disconnected

Let $X$ be a connected metric space ( with more than one point ) and $A \subseteq X$ be not closed in $X$ and such that $X \setminus A$ is not connected ; then is it true that $X \setminus B$ is also ...
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29 views

Distance geometry and pythagorean theory. Pairwise distances to absolute 2D coordinates

I don't have sufficient mathematical background. I am trying to get the absolute 2D coordinates from the pairwise comparison distances: What I have distances between points: p1-p2 = 0.3 p1-p3 = ...
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32 views

Proving that $d(a,b)=p^{-n}$ is a metric for $\mathbb{Q}$

I have the following task: If we have the metric $d:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{R}$, so that $d(a,a)=0$ and $d(a,b)=p^{-n}$ always when $a-b=p^nh/k$, where ...
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Why is the metric on $\mathbb{N}$ defined as the following?

This is from Muscat's Functional Analysis:http://staff.um.edu.mt/jmus1/metrics.pdf Show that $d(m,n) = |\dfrac{1}{m} - \dfrac{1}{n}|$, $m,n \in \mathbb{N}$ So the first two properties of the metric ...
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Let $(X,d)$ be a metric space and $Y\subset X$. Suppose $G\subset X$ is open; show that $G\cap Y$ is open in $(Y,d)$.

Let $(X,d)$ be a metric space and $Y\subset X$. Suppose $G\subset X$ is open; show that $G\cap Y$ is open in $(Y,d)$. I'm not sure how to show this result. Any solutions/hints are greatly ...
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79 views

Does every connected metric space $X$ contains a connected subset $A$ such that $X \setminus A$ is infinite?

Convention : Whenever we are going to talk about connected spaces , we will mean with more than one point . I am trying to see whether every connected metric space $X$ contains a connected subset ...
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Existence of a special kind of continuous injective function $f\colon A \to \mathbb R$, where $A$ is countable, relating to connectedness

Let $A \subseteq \mathbb R$ be a countable set ($A$ induced with usual subspace topology), then does there necessarily exist a continuous injective function $f\colon A \to \mathbb R$ such that for ...
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1answer
30 views

How to prove $(0,1) \times \mathbb{R} \, , \, (0,2) \times \mathbb{R}$ are not isometric?

I am trying to prove in an elementary way that $X_1=(0,1) \times \mathbb{R} \, , \, X_2=(0,2) \times \mathbb{R}$ (with the standrad euclidean metric inherited from $\mathbb{R}^2$) are not isometric as ...
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Is the boundary of every compact connected subset ( with more than one point ) of $\mathbb R^2$ is Homeomorphic with $S^1$ ?

Is it true that the boundary of every compact connected subset ( with more than one point ) of $\mathbb R^2$ is Homeomorphic with $S^1$ ? I was thinking that union of two closed balls touching ...
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27 views

Isotropic Metric on $\mathbb{R}^{\mathbb{N}}$

I don't know if the term isotropic is correct in this context, but I was wondering if there exists a non trivial metric $\rho $ in $X=\mathbb{R}^{\mathbb{N}}$ such that $$\forall i \in \mathbb{N} ...
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45 views

Topologically-equivalent/metrically-equivalent metrics and the same topology

Definition: Metrics $d_1$ and $d_2$ on $X$ are topologically equivalent iff $d_1(x,x_n)\to 0 \iff d_2(x,x_n)\to 0$ for every $\{x_n\}\subset X$ and $x\in X$. Definition: Metrics $d_1$ and $d_2$ on ...
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50 views

about shape of open ball in metric space

consider following shape in plane : now we have definition of open ball in every metric space : $$B_r(x_0) := \{x \in X : d(x_0,x)<r\}$$ that radius is $r$ and center is $x_0$ , my questions are ...
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38 views

I want to prove that $f$ is continuous if its graph is closed

This is an exercise from Rudin's 'Functional Analysis': Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph $f:X\to K$ is a closed subset of $X\times K$. Prove that ...
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33 views

Open and connected set in metric space [duplicate]

In a normed space, we know that if a set is open and connected, it is path connected. Is it true for general metric space or general topological space?
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distance between a real and R\Q

Please how to prove that $d(x, R\setminus Q)=0, d(x,Q)=0$ for all $x\in \mathbb{R}$ ? I know that $d(x,R\setminus Q)=\inf_{a\in R\setminus Q} d(x, a)$ but how to continue ? Can i say that As ...
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23 views

Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n (q_k-p_k)^2]^{1\over 2}$.

Let $p=(p_1,p_2,...,p_n)$ and $q=(q_1,q_2,...,q_n)$ be points in $\mathbb{R^n}$ with $p_k<q_k$ for each $k$. Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n ...
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32 views

Zero-dimensional separable metric spaces

I have to prove that every separable metric space, which is zero-dimensional is isomorphic to a closed subset of the Baire space. Maybe I can use the Baire category theorem, but I don't know how.
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21 views

How to show $dist(p,S) = 0$, then there exists a sequence in $S$ converging to $p$

Let $S \subset (M, d)$, where $(M,d)$ is a metric space Let $dist(p,S) \equiv \inf\{d(p,s) | s \in S, p \in M\}$ I wish to show that if $dist(p,S) = 0$, then there exists a $(p_n)$ in $S$ converging ...
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1answer
25 views

Type of convergence of a Cauchy sequence of functions on a complete metric space?

Let $\{f_n\}$ be a Cauchy sequence of functions defined on a complete metric space $E$. Then $f_n \to f$ on $E$. What is the type of this convergence? Is it pointwise?
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1answer
71 views

Let $f$ be a function $f:[0,1] \to [0,1] \times [0,1]$ now can we find $f$ with following conditions?

Let $f$ is a function $f :[0,1] \to [0,1] ×[0,1] $ now can we find $f$ with following conditions ?: 1- f be continues and one to one . 2- f be continues and onto . 3- f be continues and one to one ...
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59 views

Is a Normed Vector Space Necessary to Prove Path Connectedness?

Path connectedness seems to be defined in a topological space, but can the existence of a path be proven without using the functions of vector addition, scalar multiplication and norm ? For example, ...
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34 views

Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$

Let $(M,d)$ be a metric space and define: $d' : M$x$M \rightarrow R$ Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$ I know that $d'$ defines a metric on M, since d is a ...
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Does nonexpansive property in H-norm imply nonexpansive in 2-norm?

Suppose $\|f(x) - f(y)\|_H \le \|x - y\|_H$. In other words, $f$ is nonexpansive in the norm with respect to positive definite H: $\|z\|_H = z^T H z$. Can we then say something along these lines: ...
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Is the set of Darboux integrable function a metric space under the given distance definition?

Let $X$ be the set of all Darboux integrable functions in the domain $[0,1]$. Distance between two function is defined to be $$d(f,g) = \int_0^1|f-g|\,dx.$$ Is this a metric space? I ...
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1answer
23 views

Am I making some mistake in proving that $S$ is dense subset of $C[0,1]$?

Consider the space $X=C[0,1]$ with its usual 'sup-norm' topology.Let $$S= \{ f \in X : \int_{0}^{1} f(t) dt \neq 0\}$$ Show that $S$ is dense in $X$ We note that convergence with sup norm is ...
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29 views

Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n\times n$ matrices with real entries

Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n \times n$ matrices with real entries. Definition 1.26. Let $X$ be a linear space (over $R$). A ...
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Retracts and dense $G_{\delta}$ sets

Suppose $X$ is a complete separable metric space and $Y\subset X$ is a retract of $X$ with retraction $r\colon X\longrightarrow Y$. I am interested in the following question: Given a dense ...
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Is the metric on the circle, induced from the plane, not a flat one?

My question concerns the highlighted part posted below, from Wikipedia article. (Link to the revision at the time of this post.) I'd say I can't detect the curvature of the unit circle if I go along ...
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Get a bounded metric from a metric - triangle inequality for $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ [duplicate]

This is related to Proof that every metric space is homeomorphic to a bounded metric space but I can remember that if $d$ is a metric, then $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ is also a metric that ...
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1answer
16 views

Non-separability of normed spaces

I would like some hints to decide when a normed space is separable or not. I really understood the definition and the classic examples of separable spaces but when I go to show that a space is ...
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1answer
29 views

To construct a continuous map on compact metric spaces (cf. Gromov-Hausdorff Limit)

This is about Gromov Hausdorff limit on compact metric spaces (Reference A course in metric geometry - Burago Burago and Ivanov, 268p. EXE 7.5.8) Definition : $d_{GH}(X,Y)<\epsilon $ if there ...
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Compactness Theorem (Propositional Logic) and Compactness (Metric spaces). [duplicate]

Definition. A subset $E$ of a metric space $(X,\tau)$ is compact if every open cover of $E$ has a finite subcover. Theorem (Compactness Theorem). A set $\Gamma$ of formulas is ...
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infinite subset of discrete metric space is not compact

The question is Im not really sure how to go about this So far i am trying to show that for an open cover of the infinite subset X, there isn't a finite sub cover and therefore X is not compact I ...
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40 views

Metric spaces and normed vector spaces

Studying I learned that there are some theorems and definitions that need a metric structure on the space in which we are working, for example the definition of local maximum needs a metric space or ...
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97 views

Why pseudo-Riemannian metric cannot define a topology?

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. Does this imply that in cosmology, say through FLRW metric, ...
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Meaning of amalgamated metric sum of $A_n$’s over $0$ and $d_n$ inherited from $\mathbb{R}^2$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
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1answer
32 views

What can you say about interior points of a non empty subset of real numbers?

Given that A is a non-empty subset of real numbers, if I(A) denotes the set of interior points of A; then I (A) is:- a) empty. b) singleton. c) a finite set containing more than one element. d) ...
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How calculate with Riemannian metrics (e.g. Multiplication and Divison)?

I have no idea how to handle the following Riemannian metrics, how to find the estimates for the bound and how to actually calculate with $g$ and $d$. Do I need to use the matrix representation? Or ...
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1answer
15 views

Proving that a set is closed with respect to a defined metric

Let $M = [0,1]^{[0,1]}$ Prove that the set of increasing functions $$ J := \{f \in M : \forall \space a,b \in [0,1], a \leq b : f(b) − f(a) \geq 0 \} $$ is a $d$-closed subset of $M$ where ...
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compact set in a space of functions continuous in $\mathbb{R}$

As it is known the set $\mathcal{B}=\{ f:\mathbb{R}\rightarrow \mathbb{R} : f \mbox{ is continuous}\}$ it is not a metric space with the metric $d(f,g)=sup_{x\in \mathbb{R}}\| f(x)-g(x)\|$ it can ...
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1answer
49 views

Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space?

Suppose $<M,d>$ is a metric space. Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space? Or maybe lim$_{a \rightarrow b } \space d(a,b) \neq 0 $ implies ...
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2answers
48 views

Show that mapping is a contraction?

Show that the mapping $f:\Bbb R \to \Bbb R $, $f(x)=1-xe^x$ is a contraction. I tried everything i could think of but i cant get it to work. Witch is not much since i couldn't really find any ...
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3answers
42 views

Show that the collection of open balls in two metric spaces are identical

I am having trouble trying to prove the following statement. I can see why it would be true intuitively, however, I am having trouble formalising the proof as the notation is quite confusing. Show ...
2
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49 views

Limit of Riemannian manifolds is not Riemannian

I want to prove that $D$, standard unit ball in ${\bf R}^2$ with $|\ |$, with a metric $\| \ \|_1$ is a limit of Riemannian manifolds $X_i$. Here problem is to find $X_i$ (If necessary, all metrics ...
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14 views

The name of a polygon defined by multiple overlapping annuli

I am working on a problem in a metric space where points are partitioned into various annuli. If there exists multiple annuli that define a set of points then a polygon can be formed from their ...
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48 views

lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$ in Metric Space - Implications

A Metric Space $<M,d>$ is given by the Metric $M$ and distance function $d$ If there exists a Cauchy Sequence $x_n$ such that: lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$, for some $a \in ...