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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Polynomial root (using contraction mapping principle)

I am asked to provide an iterative algorithm which would lead to finding a real root of this polynomial: $$6x^5-x^3+6x-6=0$$ It is required to rely on the contraction mapping principle and Banach ...
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393 views

Trivial Metric Space

Can someone show me how to prove that the trivial metric space is indeed a metric space (if $a=b$ then $d(a,b)=0$ and if $a \ne b$ then $d(a,b)=1$)? I'm having trouble with the triangle inequality ...
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381 views

Open and closed subspace of metric space

I would be grateful for some guidance on this particular problem. Let $S=\left\{1-\dfrac1n \mid n \in \mathbb N\right\}$ be viewed as a subspace of $\mathbb R$ with the usual metric. i) Is $S$ open? ...
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Is $\mathbb{Q}^2$ homeomorphic to $\mathbb{Q}^2\setminus \{0\}$?

I know that $\mathbb{R}^2$ and $\mathbb{R}^2\setminus\{(0,0)\}$ are not homeomorphic. (For examle $\pi_1(\mathbb{R}^2)=\{e\}$, but $\pi_1(\mathbb{R}^2\setminus\{(0,0)\})=\mathbb{Z}$). But what can ...
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How to show the intersection of arbitrary compact sets is compact in a general metric space?

I understand that if you are working in $\mathbb{R}^n$, then the intersection of an arbitrary collection of compact sets is compact because it is closed and bounded. But what if you are not in ...
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4answers
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Show that the closure of a subset is bounded if the subset is bounded

Let $A$ be a subset of $X$, and let $A$ be bounded. I.e.: $\exists x_0\in X : d(x,x_0)\le K, \forall x\in A$ I want to show that $\overline{A}$, the closure of $A$ is bounded as well, but as simple ...
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A subset of a metric space is closed iff its intersection with every compact subset is closed

I want to show that a subset of a metric space $X$ is closed iff its intersection with every compact subset of $X$ is closed
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1answer
77 views

Balls in a separable metric space

Let $X$ be a separable metric space and let $(r_i)$ be a sequence of positive reals. Denote by $(x_n)$ a countable dense subset of $X$. Is it true that $X=\bigcup_{n=1}^\infty K(x_n, r_n)$?
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Metrizability is a topological property?

How could I show that metrizability is a topological property? Well, this means that if I have a set X that is metrizable and a homeomorphic function f from X to Y, then I need to show that Y is ...
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1answer
124 views

How to prove inverse metric is a metric in Y

I just need to know if my answer is right based on the following question: Let $(X,d)$ be a metric space, and let $f:X\to Y$ be a homeomorphism of $X$ onto a topological space $Y$. Define a metric ...
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Can a collection of points be recovered from its multiset of distances?

Consider $n$ distinct points $x_1,\dots,x_n$ on $\mathbb{R}$. Associated to these points is the multiset of all distances $d(x_i,x_j)$ between two points. Suppose one is only handed this multiset (you ...
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293 views

Metric completion of field of fractions

The integers have as a field of fractions the rational numbers which have a metric completion as the real numbers. The reals can be represented by infinite decimal expansions which can be approximated ...
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4answers
207 views

Definition of a metric

I'm having a hard time understanding what the definition of a metric is. From what I think I understand, it's just a method of measurement between $2$ points in $\mathbb R^n$? Is that somewhere along ...
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1answer
101 views

Prove $\operatorname{dist}(\overline{A},\overline{B}) = \operatorname{dist}(A, B)$

This is the last question on the exercise sheet and I am having real trouble formalizing my intuitions. It should be obvious. Since the closure of a set is the set of all points in the universe with ...
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94 views

How does one show that there exists some $z \in X$ such that $f(z) = z$ under certain circumstances?

In a previous exercise, one was asked to show that the sequence $(x_n)_{n > 0}$ in $X$ (with $(X,d)$ a non-empty, complete metric space) in which we have $d(x_n,x_{n+1}) \leq \theta d(x_{n-1} , x_n ...
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Are these subsets of $\mathbb{R}$ homeomorphic?

Consider the following subspaces of $\mathbb{R}$ with the usual topology: $$X = (0, 1) \cup \{2\} \cup (3, 4) \cup \{5\} \cup \cdots \cup (3n, 3n + 1) \cup \{3n + 2\} \cup\cdots$$ $$Y = (0, 1] \cup ...
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229 views

Second Countability of Euclidean Spaces

Sorry I know this is a stupid question. However I got stuck on this for quite a while. I'm trying to prove that Euclidean spaces have a countable base, which can be constructed by taking all the open ...
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51 views

A matching distance

Consider $\sigma,\sigma' \in \{1,\dots,p\}^n$, and let $$ d(\sigma,\sigma') = \min_{\pi \in S_p} \frac1n \sum_{i=1}^n 1_{\{ \pi(\sigma_i) \neq \sigma'_i \}} $$ where $S_p$ is the symmetric group on ...
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Open sets in a topology produced by metrics.

For each $i\in I$, $$d_i:X^2\to \Bbb R$$ is a metric and $\mathcal T$ is the coarsest topology on $X$ containing all topologies produced by the $d_i$ . For $U\subseteq X$ we have: $$\forall x \in ...
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$\sqrt{26}$ : Find by bisection method

How would you find the root of $\sqrt{26}$ by bisection method? A step by step solution would be greatly appreciated!
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1answer
545 views

A closed set in a metric space is $G_\delta$

How do I prove that a closed set $F$ in the metric $(X,d)$ is $G_\delta$. Let $n\in \mathbb{N}$. I consider $B_n={F}=\bigcup_ {x\in F} B(x,{1\over n})$, which is a collection of an open ball. Then ...
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1answer
224 views

If $ x $ is a limit point of a Cauchy sequence $ (x_{n})_{n \in \mathbb{N}} $, then $ (x_{n})_{n \in \mathbb{N}} $ converges to $ x $.

Define a point $ x $ in a metric space $ X $ to be a limit point of a sequence $ (x_{n})_{n \in \mathbb{N}} $ if there exists some subsequence $ (x_{n_{k}})_{k \in \mathbb{N}} $ of $ (x_{n})_{n \in ...
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If $x$ is not in $A$, a closed set in a Metric space then $d(x,A)>0$

If $A$ is a closed in a metric space $(X,d)$ with $x\notin A$, I need to show that $d(x,A)>0$. Now assume $d(x,A)=0$ then $\exists x_n\in A $ s.t.$d(x_n,A)=0$ then there is a sequence in $A$ s.t. ...
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225 views

Inverse of Heine–Cantor theorem

We have by Heine–Cantor theorem that: If $M$ and $N$ are metric spaces and $M$ is compact then every continuous function $f : M \to N$, is uniformly continuous. Is the inverse of this theorem ...
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109 views

Show that $\overline{\overline{X}} = \overline{X}$ for all subsets $X$ of a metric space $S$.

I have to prove this and I am not sure how to go about it. Help please
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109 views

general topology exercise

Consider $\mathbb R^n$ with the usual metric. Let $U \subset \mathbb R^n$ be an open set and $K \subset U$ a compact set. Is the following affirmation true? There exists an open set $U^{'}$ in ...
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776 views

Understanding Complete Metric Spaces and Cauchy Sequences

From my own definition, I have concluded that a complete metric space is a set and a metric where the set consists of no holes in it. Book definitions describe that "A complete metric space is a ...
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3answers
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A nbhood of an accumulation point contains infinitely many points

Let $A$ be a set, $X$ a metric space, $x$ an accumulation point of $A$ that is, every nbhood of $x$ contains a point $a \in A$, $a \neq x$. I wrote a proof of the fact "every nbhood of $x$ contains ...
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convergence of functions on probability measure

I am studying a problem in game theory, but I am lacking on knowledge to deal with a continuum of distribution functions convergence. $\mathfrak{F}([0,1])$ is the set of distribution functions over ...
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What is the relation between convex metric spaces and convex sets?

Here's another question that came to mind when I was reading the article on convex metric spaces in Wikipedia: According to the article, "a circle, with the distance between two points measured along ...
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Metric of space of plane curve

I am looking for a metric $d$ for smooth 2D curves. Hence $d(x,y)$ is the distance between the curves x and y. For the moment, we may assume that $x$ and $y$ are just directed line segments. Do you ...
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1answer
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The space of continuous, bounded functions from a metric space $X$ to $\mathbb R$

Let $(X,d)$ be a metric space. We denote by $C_b(X;\mathbb{R})$ the space of continuous and bounded functions from $X$ into $\mathbb{R}$, equipped with the sup-norm metric. We define a mapping $O: X ...
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Why is Hausdorff metric defined this way? [duplicate]

From Wikipedia The definition of the Hausdorff distance can be derived by a series of natural extensions of the distance function $d(x, y)$ in the underlying metric space $M$, as follows:[4] ...
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Critique this proof on compactness.

Problem: Prove or disprove, the metric space $X$ containing infinitely many points with the discrete metric is compact. Write a proof in the language of sequences and covers Proof: Take $(1/n) \to ...
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Show that $\rho(x,y) = |\sin(x)-\sin(y)|$ is not a metric on $\mathbb{R}$?

Show that $\rho(x,y) = |\sin(x)-\sin(y)|$ is not a metric on $\mathbb{R}$ and in what condition must be imposed on a function $f:\mathbb{R}\to\mathbb{R}$ in order for $\rho(x,y)=|f(x)-f(y)|$ to be a ...
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Show that there is a compact neighbourhood $B$ of $x$ such that $B \cap F = \emptyset$.

Let $X$ be a compact Hausdorff space, $F \subset X$ closed and $x \notin F$ . Show that there is a compact neighbourhood $B$ of $x$ such that $B \cap F = \emptyset$. I'm trying to use the fact that ...
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1answer
58 views

Show that for each $a\in [0, \infty)$, the subspace $[a, \infty)$ is also compact

i) Show that the intervals $(a, \infty)$, $a \in (0, \infty)$ together with $\emptyset$ and $[0, \infty)$ form a topology on $[0, \infty)$. ii) Show that in this topology $[0, \infty)$ is compact. ...
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$\operatorname{dist}(A, B) = 0 \land A \cap B = \emptyset \implies \partial A \cap \partial B \neq \emptyset$

A similar question: Distance of two sets and their closest points The question above, however, defines distance differently. The definition we work under is: $$\operatorname{dist}(A, ...
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Understanding a claim — which interpretation is right?

I'm trying to disprove the following claim. If $A$, as a subspace of $X$, has discrete topology, then $X$ has discrete topology. The statement ...
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1answer
159 views

Metrics with infinite distances.

I've been wondering about the spaces $\Bbb R\cup\{-\infty,+\infty\}$ and $\Bbb C\cup\{\infty\}.$ Is there a useful generalization of the definition of a metric they satisfy? I thought it would be ...
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1answer
88 views

A metric such that $d(0,1)>1000d(0,2)$

I need to find a metric on $[0,2]$ such that $d(0,1)>1000d(0,2)$ Here is the the example I came up with the following $d(x,y)=\begin{cases} 0\ \ x=y \\ 1 \ \ x\neq y+1\\ 1001 \ \ x=y+1 ...
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Proof is contradicting what I know to be true, what is wrong?

I'm trying to prove that a given function is a metric on some set. I'm confused now because the maths is not not adding up. Let $X=${$x \in \mathbb{R^2}:\lvert x \rvert =1$}. Given $x,y \in X$, ...
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658 views

Does continuous convergence imply uniform convergence?

Question Related to a nice problem I met yesterday, a question arises: Suppose $\{f_n\}$ is a sequence of mappings from a connected complete metric space $X$ to a metric space $Y$. Given $f\colon ...
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185 views

Why is a rectangle not a neighborhood of its corners?

I'm trying to puzzle out a statement given in the Wikipedia article on topological neighborhoods, which uses this definition: If $X$ is a topological space and $p$ is a point in $X$, a ...
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160 views

Finding an isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$

Prove the isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$, where $B$ is the collection of all bi-linear mappings. In order to do so, give a natural isomorphism between ...
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1answer
210 views

How do I sketch the following metrics:

In $\mathbb{R}^2$ sketch $B((1,2),3)$, the open ball of radius $3$ at the point $(1,2)$, with the following metrics: a.) the post-office metric given by $$d(x,y) = \left\{ \begin{array}{l l} ...
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Does a norm have to map to $\mathbb R$?

Wikipedia defines a norm on a vector space $V$ as a function $p : V \mapsto \mathbb R$. I've seen this defined similarly elsewhere. However, it seems to me that a real codomain isn't always necessary. ...
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Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric. [duplicate]

where $(X,d)$ is a metric and $x,y \in X$. I know we need to show: non-negativity: $d(x,y)\geq$ 0 $d(x,y)=0$ if and only if $x=y$ symmetry: $d(x,y)=d(y,x)$ $d(x,z)\leq d(x,y) + d(y,z)$ I think we ...
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How to show that a point is not an interior point?

I understand that in order to show that a point, $x$, is an interior point of some set $A \subset B$, where $(B,d)$ is a metric space you just need to show that you can have an open ball around $x$ ...
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184 views

If $f_n(x)=x^n$ converges to $f$, why is $f$ not continuous?

I was reading my Analysis course notes and had some trouble. I hope you can help me. Let $C(X)=\{ f | f:X \longrightarrow \mathbb{R} \text{ is a continuous function}\}$. It was already stated and ...