# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
### 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 ...