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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A question on countability of isolated points of a subset of R

The question is to prove that with respect to the euclidean metric on the Real numbers prove that if A is any subset of R, isoA is countable and hence deduce that if A is uncountable the A' is ...
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3answers
693 views

Closure and limit of a sequence

Let $E$ be a subset of a metric space $(S,d)$. I'm trying to show that an element is in $\overline{E}$ if and only if it is the limit of some sequence of points in $E$. Suppose there is a sequence ...
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1answer
71 views

Is it true that bounded metric can never be induced by norm.

Let $(X, d)$ be a metric space where, $d$ is metric on $X$. We know that metric space $X$ is called bounded if there exists some number $r$, such that $d(x,y) ≤ r$ for all $x$and $y$ in $X$. I want ...
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1answer
208 views

Cauchy filters in metric spaces

Some terminology: Let $(X,d)$ be a metric space. A filter $\mathcal F \subseteq \mathcal P (X)$ is Cauchy if $\forall \epsilon >0 \exists x\in X: B_\epsilon(x)\in \mathcal F$. A filter ...
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1answer
302 views

Boundary and closure of a set

I'm trying to show that a point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement. Let $\overline{E}$ denote the closure of $E$ and $E^\circ$ be the ...
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1answer
87 views

Is a Borel subset in an analytic subset of a Polish space still analytic?

I encounter some problem like this Assume $A$ is a Borel subset of $B$. $B$ is an analytic subset of a Polish space $C$. Is $A$ an analytic set in $C$? while reading a book. But I don't know the ...
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4answers
186 views

A 2nd countable metric space is separable

A 2nd countable metric space is separable Can anyone give an idea on how to prove that, I could prove the converse quite easily but I am not sure which set to consider in order to show that it is ...
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2answers
95 views

Is $\tau(x,y)\leq\sqrt{n}d(x,y)$ where $d$ is the Euclidean metric, and $\tau$ the taxicab metric?

Suppose $d$ is the Euclidean metric on $\mathbb{R}^n$, $d(x,y)=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}$, and $\tau(x,y)=\sum_{i=1}^n|x_i-y_i|$ is the taxicab metric on $\mathbb{R}^n$. When showing this two ...
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4answers
513 views

Prove if the set is closed? Bounded? Compact?

Consider the metric space $(\mathbb{Q},d)$ where $\mathbb{Q}$ denotes the rational numbers and $d(x,y)=|x-y|$. Let $$E:=\{x \in\mathbb{Q}:x>0, 2<x^2<3\}$$ Is $E$ closed and bounded in ...
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49 views

If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$, does it follow that $A$ is dense?

If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$ then $A$ is dense in $X$, right? Once I did one problem which says $d(x,A)=0\Leftrightarrow x\in \bar{A}$ so by the condition here we get ...
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1answer
200 views

Continuity of $d(x,A)$

I am doing a head-check here. I keep seeing this theorem quoted as requiring $A$ to be closed (as in Is the function distance continuous?), but I don't think that it is needed. Theorem. Let ...
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2answers
210 views

Finding an example of a bounded sequence in a complete metric space such that the sequence has no partial limit

I'm working through an analysis text and I've come across this exercise: Give an example of a complete metric space $X$ and a bounded sequence $\left(x_{n}\right)$ in $X$ such that the sequence ...
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2answers
65 views

Any practical difference between the metrics $d_1=\sup\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ and $d_2=\max\{\left|x_j-y_j\right|:j=1,2,…k\}$?

I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for which I've already done a ...
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624 views

Metric assuming the value infinity

If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces? The reason I ask is that I saw this theorem: Given a finite measure space ...
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1answer
79 views

Intuition behind closed subsets of a metric space?

Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space. Consider a metric space $$(X,d)$$ Then consider a subset of this space$$F$$ What the book ...
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1answer
73 views

Continuity in metric space

Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by $$ F(x,y) = \left( x^3 y,\ \ln(x^2 + y^2 + 1),\ \cos(x - y^2) \right) $$ When trying to show why $F$ is continuous where should I start?
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265 views

Compact metric space group $Iso(X,d)$ is also compact

Could you tell me how to prove that if metric space $(X,d)$ is compact, then the group $Iso(X,d)$ is also compact? The group $Iso(X,d)$ is considered with topology determined by a metric $\rho$ on ...
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1answer
145 views

All closed balls are compact each isometry is bijective

Let $(X,d)$ be a metric space in which all closed balls are compact and such that for any two points $x,y \in X$ there exists a function $u \in Iso(X,d)$ such that $u(x)=y$. Prove that then each ...
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100 views

Question on Contractions

Let $S = \{x \in \mathbb{R}^n ; \|x\| \le 1 \}$ and $f: S \to S$ be a contraction. Determine one can have $f(S) = S$. I really need some help with this question. In advance I wanted to give all ...
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1answer
339 views

Uniformly continuous function - modulus of continuity

Give an example of a uniformly continuous function $f: (X,d) \rightarrow (Y,\rho)$ for which there doesn't exists a modulus of continuity $\omega: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that: ...
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1answer
2k views

Every subsequence of $x_n$ has a further subsequence which converges to $x$.Then the sequence $x_n$ converges to $x$.

Is the following is true? Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. I ...
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1answer
103 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
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1answer
92 views

Square matrix $\|Ax-Ay\|\le \|x-y\|$

Could you give me an example of a square matrix $A\in \mathcal{M}_{2 \times 2}$ or $\mathcal{M}_{3 \times 3}$ for which we have $\|Ax-Ay\|\le \|x-y\|$, $ \ \ x, y \in \{0, e_1, . . . , e_n\}, \ \ e_1, ...
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2answers
805 views

Proving that Euclidean space having the infinity metric is a complete metric space (stuck)

I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space. I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
3
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3answers
68 views

Prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ is a metric on $\mathbb{R}^k$

I am trying to prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ is a metric on $\mathbb{R}^k$. So far I know that $\text{dist}\left({x,y}\right) \leq ...
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Is any compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?

Every compact metric totally disconnected perfect space is homeomorphic to a Cantor space. Is every compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space? ...
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101 views

Definition of open set/metric space

On Proof Wiki, the definition of an open set is stated as Let $(M,d)$ be a metric space and let $U\subset M$, then $U$ is open iff for all $y\in U$, there exists $\epsilon \in \mathbb{R}_{>0}$ ...
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1answer
130 views

How to show C_e is closed and not dense in C.

Let $C_{e}([-1,1],\mathbb{R})$ denote the set of even functions in $C([-1,1],\mathbb{R})$ (a) Show $C_e$ is closed and not dense in $C$. (b) show the even polynomials are dense in $C_e$, but ...
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Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$

From numerical test, I know $x=1$ is an attractive fixed point of the function $$ f(x)=\frac12 \left(x+\frac{1}{x}\right), $$ on $(0,\infty)$. Is there a way to prove it? Since $$ ...
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1answer
261 views

Metric and the triangle inequality

Let $A,B$ be finite subsets of the natural numbers. If we let $d(A,B)=\sum_{x\in A\mathbin\Delta B} 2^{-x}$, where $A\mathbin\Delta B=(A\cup B)\setminus (A\cap B)$ is the symmetric difference between ...
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0answers
41 views

A sequence of embedded closed balls that have empty intersection [duplicate]

I'm reading soviet textbook "Elements of theory of functions and functional analysis" by Kolmogorov and Fomin. There is an exercise is in it: show example of complete metric space and a sequence of ...
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1answer
101 views

Hausdorff distance between metric spaces

I started to read a bit Gromov's paper 'Metric Structures for Riemannian and Non-riemannian Spaces'. Tthe Hausdorf distance of two metric spaces $X$ and $Y$ is defined by using isometric embeddings ...
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1answer
228 views

drawing open balls for the radar screen metric

What do the unit balls of this metric look like for radii 1/2, 1 and 3/2 please ? Radar screen metric is $d(x,y):= \min(1, \|x - y\|_2)$, where the subscript $2$ after the last $\|$ indicates it's ...
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1answer
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For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$ [duplicate]

I was thinking about the following problem: For two disjoint compact subsets $A$ and $B$ of a metric space $(X,d)$ show that $d(A,B)>0.$ I'm having doubt with my attemp. Please have a look and ...
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164 views

Ergodic theory question about the support of a measure.

I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question: ...
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1answer
134 views

How to show that$(X,d)$ is totally bounded

How to show that a metric space $(X,d)$ is totally bounded $\iff$ every infinite subsets of $X$ contains distinct points which distinct points that arbitrarily close to each other. I don't know how ...
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1answer
87 views

$X \cong Y$, $X$ complete $\implies Y$ complete? [duplicate]

Let $\cong$ denote the homeomorphic notation. Let $X,Y$ be metric spaces, and let $X \cong Y$. If $X$ is a complete metric space does it imply $Y$ is also complete.
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How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded? [duplicate]

Let $X$ be a non-compact metric space. How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded?
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546 views

Is my proof correct? (minimal distance between compact sets)

I'm working out the following problem form Ahlfors' Complex Analysis text: "Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that ...
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1answer
210 views

(Non) Complete, (Non) separable, (Non) metrizable spaces

After studying several kinds of topological spaces (Like $L_p, C[0,1]$) etc., I thought it would be useful to me (and to others) if I tabulated some of them under 3 categories: Completeness(C), ...
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1answer
46 views

Is $D^n$ defined when $n = 0$?

Define $D^n = \{x \in \mathbb{R}^n : |x| \le 1\}$. Is $D^n$ defined when $n = 0$? I would say no, since I don't think $\mathbb{R}^0$ is defined.
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Intersection of open balls in a metric space

I am wondering about the following question: Given a (countable) sequence of nested open balls: $$ B_1 \supseteq B_2 \supseteq \cdots $$ Not necessarily having the same same center. All having ...
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1answer
240 views

Calculate the distance between the points $(1, 2, \dots, n)$ and $(2, 3, \dots, n, 1)$

I know that the operation to find the distance between two vectors is: $$\sqrt{(b_1-a_1)^2+(b_2-a_2)^2+...+(b_n-a_n)^2}.$$ So the distance between $(7, 5, 3, 1)$ and $(1, 3, 5, 7)$ is: ...
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1answer
37 views

Metric spaces and curvature

Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
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1answer
284 views

Do the bounded sequences in any metric space form a complete metric space?

I know that the set of all bounded sequences over $\mathbb R$ is complete w.r.t. sup norm. Similarly the set of all bounded sequences over $\mathbb C$ is complete w.r.t. sup norm. Does this ...
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1answer
98 views

Coordinate transform

Can anyone see what transformation $$r\to f(r)$$ transforms $$\exp(2\phi(r))(dr^2+r^2d\theta^2)$$ to $$df^2+\sinh^2(f)d\theta^2$$? I there a systematic way to attack such a problem -- rather than just ...
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25 views

A better way to see this relation concerning Ricci tensor components

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
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110 views

Metric spaces and distance functions.

I need to provide an example of a space of points X and a distance function d, such that the following properties hold: X has a countable dense subset X is uncountably infinite and has only one ...
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2answers
54 views

Which step fails if we would assume $F=(a,b) \subset ℝ$ in the Heine-Borel theorem

I have a question about the theorem: "Every k-cell in $ℝ^k$ is compact". I think it is quite a hard proof, and when I was thinking about it I don't understand the following: Suppose $F$ is not ...
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69 views

Showing a Particular Function Between Two Metric Spaces is Continuous

$\fbox{Hypothesis}$ Suppose $(M,d)$ is embedded densely into the two complete metric spaces $(M', d')$ and $(M'', d'')$. Suppose we define $\rho: M \rightarrow M''$ as the identity mapping. ...