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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are two metrics with same compact sets topologically equivalent?

are two metrics with same compact sets topologically equivalent ? I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set is ...
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Show that $|d(m,n) -d(n,o) | \leq d(m,o)$ for a metric space

Problem Let $(M,d)$ be a metric space. Show that $$|d(m,n) - d(n,o)| \leq d(m,o) \ \forall m,n,o \in M$$ Since $(M,d)$ is a metric space I know it fufills the triangle inequality. So if I ...
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Proving that union of epsilon nets is an epsilon net

While reading a paper, I came across these definitions and claims: Definition: Given $p \in \mathbb R^d$, and $H$, a set of hyperplanes, let $\text{Violate}_p(H) = \{h \in H: h \text{ is strictly ...
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57 views

Is $\mathbb{R}$ an open ball in $\mathbb{R}$?

If we write $B(0,\infty)$ as the open ball then $\mathbb{R}$ is an open ball in $\mathbb{R}$. Is it correct?
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In a complement of a closed set find a point that is closest to any point in the closed set

Is it true that for $U$ closed and nonempty in a metric space $X$, let $a\in X\setminus U$, then there exists a $b\in U$ such that $d(a,b)\leq d(a,x) \forall x\in U$? I think it is correct because it ...
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56 views

Is there anything wrong in the following proof?

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces and let $U\subseteq X$ and $V\subseteq Y$ such that $U$ and $V$ are respectively open in $X$ and $Y$. Show that $U\times V$ is open in ...
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18 views

Connectedness proof of a metric space. [on hold]

Assume that there is a $p$ in a metric space $(\chi, d)$ such that the function $f(q) = d(p, q)$, $q \in \chi$ omits the value $c > 0$, but takes values greater than $c$. Show that $(\chi, d)$ is ...
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20 views

An example of the set of distances of two points in two different closed sets having no infimum

On a problem set for my Analysis in Several Dimensions class (basically real analysis on multivariable functions), I encountered this question: Let $(X, d)$ be a metric space, let $C ⊂ X$ be a ...
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1answer
27 views

Is every compact metric space hereditarily separable?

Let $X$ be a compact metric space. I see why all open and closed subsets of $X$ are separable. But is every subset of $X$ necessarily separable? EDIT: Since $X$ is separable metric, it embeds into ...
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How to define and compute the norm of a vector with riemannian metric?

Let us consider for example, the riemannian metric $g=e^xdx^2+dy^2$ (it is symmetric and definite positive), with associated matrix $\begin{pmatrix} e^x & 0\\ 0 & 1 \end{pmatrix}$. Consider ...
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21 views

Cauchy continuous implies standard continuity

Let $f$ be Cauchy continuous. $f$ is Cauchy continuous if for any Cauchy sequence $\{x_{n}\}$ in $(X,d_{X})$, $\{f(x_{n})\}$ is a Cauchy sequence in $(Y,d_{Y})$. Show that Cauchy continuous $\implies$ ...
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17 views

Metric spaces inside of metric spaces

Let $(X, d)$ be a metric space, $Y$ ⊂ $X$ and consider the metric space $(Y, d)$. Show that every open set $U$ in $Y$ has the form $U$ = $V$ ∩ $Y$ for an open set $V$ ⊂ $X$. Show that ...
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1answer
28 views

What is the meaning of “infinitesimal structure”?

Reading a Differential Geometry book I found this sentence: "A main theme in analysis on metric spaces is understanding the infinitesimal structure of a metric space." I cannot understand the meaning ...
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14 views

What is the max size of subset of a Hamming Space with this property

Keep in mind I know virtually no coding theory, I simply recognized this as an equivalent formulation to another question I was considering. Let $F(r,n)$ denote the set of all words of length $r$ on ...
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1answer
36 views

A metric between functions on $\mathbb{R}^2$

I want to measure the distance between functions $f$ and $g$ (not necessarily continuous) on a bounded subset $M\subset\mathbb{R}^2$. I assume $f$ and $g$ are locally integrable and bounded on $M$. ...
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$A,B$ closed subsets of $\mathbb R^n$ , when can we say (other than compact-ness of $A$ or $B$ ) $\exists b \in B$ such that $dist(A,B)=dist(b,A)$ ?

Let $A,B$ be disjoint closed subsets of $\mathbb R^n$ , when can we say ( weaker than compact-ness of $A$ or $B$ ) that there exist $b \in B$ such that $dist(A,B)=dist(b,A)$ ? I know that if $A,B$ are ...
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Hausdorff dimension of Sierpinski triangle

https://en.wikipedia.org/wiki/Hausdorff_dimension#Behaviour_under_unions_and_products Wikipedia page says that if $ \underset{i \in I}{\cup} X_i = X$ and $I$ is countable then $dim_{Haus}(X) = ...
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1answer
27 views

Does computing distance in $N$ dimensions have any application beyond $N=3$?

So I had to implement the distance formula earlier in programming, and had a thought that I should make it work for $N$ dimensions. I then smacked myself and realized that I'm programming for a ...
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1answer
16 views

$X$ is a normed linear space such that for some compact $K\subseteq X$ , $\operatorname{span} K$ is dense in $X$ then is $X$ separable?

Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and ...
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2answers
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If $X$ is a compact metric space and $E_n$ is closed nonempty subset, show that $\cap_{n=1}^\infty E_n$ is nonempty.

Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is ...
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How limiting/ heavy is the “triangle inequality” assumption?

Suppose a theorem proves something about a family of distance measures, with this the triangle inequality assumption. How limiting this assumption is in reality? What are some real-world examples of ...
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36 views

Proving $\mathbb{R}/\sim$ is homeomorphic to unit circle

Let $S$ be the unit circle in $\mathbb{C}$, standard topology. Define the equiv. rel. $\sim$ on $\mathbb{R}$ as $x\sim y\iff x - y\in\mathbb{Z}$. I would like to prove that $\mathbb{R}/\sim$ is ...
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Continuity of a function between metric spaces

I want to show: Let $(X,d)$ be a metric space and $A \subset X$ be a closed subset. Define $f: X \to \mathbb{R}$ by $$ f(x) = d(x,A) := \inf_{y\in A}d(x,y), \phantom{.} \forall x \in X.$$ Show ...
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25 views

Euclidean metric in $\mathbb{R}^n$; the singleton is not open in such a metric space

I am trying to prove this but just don't see it. We are talking about openness in the metric sense, yes? So, my attempt is Let $x \in \mathbb{R}^n$ and $d$ represent the Euclidean metric, ...
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Is $\{\frac{m}{10^n}\mid m,n\in\mathbb Z,\quad n\geq 0\}$ dense in $\mathbb R$?

The set $S$ of real numbers of the form $m/(10^n)$, $m,n$ integers and $n$ greater than equal to $0$, is dense sunset of $\mathbb R$ or not?? I know dense means closure of $S$ in $\mathbb R$ is ...
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1answer
32 views

An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and ...
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83 views

Property of Nowhere Dense Sets

I am trying to prove the following statement regarding nowhere dense sets: "In a metric space X, the frontier of an open set is the set of accumulation points of a discrete set." As far as my ...
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What simple topological properties of conic sections can be explored?

In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them? Edit: To clarify, ...
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$A \in SO(3,\mathbb R)\setminus\{I\}$ , then there are exactly two points in $S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$ which are fixed by $A$?

Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$? Or equivalently we ...
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Topological spaces that remain non-metrizable, if the definition of metric space allows $d(x,y) = 0$ where not necessarily $x = y$?

In the definition of metric space, only one thing strikes me as unnatural: the requirement that $d(x,y) = 0$ implies $x = y$. As a programmer, I don't find it uncommon to deal with equivalence ...
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61 views

Give a “constructive” proof of the fact that in a metric space the intersection of two open balls is open

Main Question Can someone give a "constructive" proof of the fact that, Let $(X,d)$ be a metric space and $x,y\in X$. Let $B_d(x,r_x)$ and $B_d(y,r_y)$ be two open balls centered respectively at ...
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Proof that a subset of metric space with euclidian norm is open iff the same subset is open in metric space with Manhattan norm

For $\mathbb{R}^2$ we have the euclidian norm $$(x_1,x_2)\mapsto\sqrt{x_1^2+x_2^2},$$ and the Manhattan norm $$(x_1,x_2)\mapsto|x_1|+|x_2|.$$ Let $d_E$ and $d_M$ be the metrics defined by these norms, ...
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1answer
49 views

Open or closed set in $\mathbb{R}$

I have this set $A=\left\{\frac{1}{n}|n\in\mathbb{N}\right\}$ I need to show that it is neither open or closed in $\mathbb{R}$. And that the union ...
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44 views

Functions between metric spaces (and how they relate to closures of sets)

Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$ we have $f(\overline{A})\subset \overline{ f(A) }$ ...
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37 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
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1answer
36 views

Preservation of completeness through a continous onto mapping

Let $(X_{1},d_{1})$ and $(X_{2},d_{2})$ be metric spaces and $f: X_{1} \to X_{2}$ be a continuous onto map such that $$ d_{1}(x,y) \leq d_{2}(f(x),f(y)) \hspace{2mm} \forall\phantom{i}x,y \in ...
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Interior of a set in a metric space

if $E$ is a metric space nd $B\neq E$ how to prove that: $$\overset{\circ}{B}=\bigcup_{n=1}^{\infty} (\{x\in E, d(x, E\setminus B)\geq \frac1n\})$$ i don't know how to start
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Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
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Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
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1answer
22 views

For $U\subseteq Y\subseteq X$, prove that $U$ is open in $Y$ iff there is a $V\subseteq X$ such that $U=Y\cap V$

Let $(X,d)$ be a metric space, with $Y$ a subset of $X$. How do I prove that a subset $U\subseteq Y$ is open in the metric space $(Y,d|_{Y\times Y})$ iff there exists an open subset $V$ of $X$ ...
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1answer
42 views

If $\sum_{n=1}^{\infty}x_n^2<\infty$ and $\sum_{m=1}^{\infty}x_n^2<\infty$, is $\sum_{k=1}^{\infty}(x_n)_k^2(x_m)_k^2<\infty$? [duplicate]

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ Prove that $l^2$ is complete with ...
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Prove that all three metrics induces the same topology on $X_1\times X_2$

Prove that if $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces on $X_1\times X_2$ and metric $d:(X_1\times X_2)\times (X_1\times X_2)\rightarrow R$ is defined in following way: ...
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59 views

How to resolve the apparent paradox resulting from two different proofs?

Definition of Open Ball Let $(X, d)$ be a metric space and let $r\in\mathbb{R}^+$. Then the set, $B_d(x, r) := \{y \in X : d(x, y) < r\}$ will be said to be the open ball of radius $r$ ...
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Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in ...
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1answer
32 views

Function between two metric spaces?

I need to come up with: two metric spaces ( X , d ) and ( Y , p ) A continuous function f: X → Y A Cauchy sequence {xn} in X that isn't mapped to a Cauchy sequence in Y My idea was to make ...
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2answers
40 views

Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
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2answers
31 views

What values of $b$ such that $f_n(x)=b\cos\left(\frac{x}{n}\right)$ converges uniformly?

For what values of $b$ does the sequence of functions: for each $n\in\mathbb{N}$, let $$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$ converge uniformly in the space $C[0,1]$ equipped ...
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0answers
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+100

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
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2answers
59 views

A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
2
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2answers
37 views

Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...