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 the triangle inequality property of a metric space important?

From my understanding, when we use metric spaces, we are trying to measure how "different" certain elements in a metric space are from one another. We all know that a metric space $(S,d)$ satisfies: ...
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In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
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Gromov-Hausdorff distance between a “Line segment” and a “Zylinder”

I want to prove the following statement: For $X= S^1 \times [0,1]$ and $Y= [0,1]$ with the intrinsic metrics one has $d_{GH}(X,Y) = \frac{\pi}{2} $ where $d_{GH}$ denotes the Gromov-Hausdorff ...
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Connected $G_\delta$ non-singleton, proper subsets in a connected complete metric space with more than one point

This is a question related to my last; I have still not solved it. Maybe this one is easier: Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a ...
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35 views

Proving $d_E$ is a Metric on $\Bbb R^n$

This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1): $d_E(\mathbf {q,p}) = 0 \iff ...
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79 views

Zer0-dimensional, countable, 1st countable T1 space is metrizable?

Show that every countable, first countable, zero-dimensional T1 space $X$ is metrizable. I know that T1 space means that all its singletons are closed. Also, zero-dimensional means that $X$ has a ...
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493 views

Closure of an open ball equal to the closed ball

If $X$ is a discrete space (metric). Then the closure of a open ball $B_1(x)=\{x\}$ is $B_1(x)=\{x\}$, and the closed ball is $X$, therefore do not coincide. You know another example such that: ...
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Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
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89 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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Statistics for random permutations

Let $S_n$ be the symmetric group on $n$ elements, let $d$ be the Cayley distance, and let $m$ be a Haar measure on $S_n$. Let $s$ denote a random permutation with respect to $m$, i.e., $s$ is an ...
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An (extended) semimetric on surfaces

Given a smooth surface $S \subseteq \mathbb{R}^3$, like the surface of sphere, we can define the following extended semimetric $d : S^2 \to [0, \infty]$, where $$ d(x,y) = \inf\{\lVert x - p\rVert + ...
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64 views

Kullback-Leibler $KL(p,q)\neq KL(q,p)$

I'm doing a course of Artificial Intelligence and in my homework I must to provide a counter example to show that the Kullback-Leibler distance is not a symmetric function of its arguments: $$ ...
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101 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
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91 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
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When is a function space a Fréchet space?

Let $Q$ be a space of indices, and let $(V, |\cdot|)$ be a Banach space of values. Define the function space $X = C(Q,V)$, and equip it with the topology generated by seminorms $\|x\|_D := \sup_{d \in ...
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Is there a non-complete and non-separate metric space?

Is there a (non-trival) non-complete and non-separate metric space? Some notions are here: math.stackexchange.com/questions/182316.
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On local rings $(R,m)$ having a metrizable $m$-adic topology

If $R$ is a local ring with maximal ideal $m$ and the intersection of powers of $m$ is $0$, then the $m$-adic topology is metrizable. Is there a condition on $R$ assuring that the metric space so ...
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44 views

The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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48 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
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Completeness of metric space induced by outer measure (similar to Nikodym metric)

Let $S_\mu$ be a semi-ring of subsets of $X$ and $\mu$ be a $\sigma$-additive measure on $S_\mu$. Let $\mu^*$ be the induced outer measure on $P(X)$. Define a relation $\sim$ on $P(X)$ by ...
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86 views

Constructing a metric for a metrizable space.

I am an electrical engineering PhD student, not a formally educated mathematician. I could prove that the main space I am using in my dissertation is metrizable (using Urysohn's metrization ...
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95 views

Show that the distance between these two sets is not bounded.

I have a homework question that asks: "Consider the curve $\gamma : [1, \infty] \to \mathbb{R}^2$ defined by $\gamma (t) = \langle t \cos (\ln t), t \sin (\ln t) \rangle$. Show that this curve is ...
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363 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
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Is this case possible (hedgehog metric, colinearity)

My topology class was asked to prove that the hedgehog metric was indeed a metric (the details are irrelevant for my question). This does not concern the proof itself, but rather the structure of the ...
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48 views

Does this have a name (metric space related measure of closeness)?

Consider a metric space $(M,d)$, and let $D: M \times M \to \mathbb{R}_+$ be a measure of similarity on it, so that $D(x,y)$ is large when $x$ and $y$ are close (i.e., $d(x,y)$ is small). Consider a ...
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277 views

Bounded Lipschitz Metric on Space of Positive Measures

The bounded Lipschitz metric ($d_{BL}$) metrizes the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ $$d(\mu, \nu) = ...
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82 views

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene $x\sigma=(x_{\sigma_n})$ where $\sigma:\mathbb N\to\mathbb N$ is strictly increasing. Then $(x_n)$ is ...
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Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
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Distance between sets in a partial metric space

How can we define distance between two sets say, $A$ and $B$ in a partial metric space $(X, p)$? Will it be non-symmetric as in the case of a metric $d$, i.e.; we have $d(A, B)$ not equal to $d(B, A)$ ...
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a problem on metric spaces

I am reading the book by Burago and Ivanov "A course in metric geometry". I tried to do some problems but have some difficulties. For example, page 66 exercise 3.1.26: Let $(X, d)$ be a metric space ...
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convergence in metric space

Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$. Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
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Regarding nowhere dense subsets and their measure.

A while ago it was made clear that a nowhere dense subset $P \subset [0;1]$ whose Lebesgue measure $\mu(P) = \mu([0;1]) = 1$ doesn't exist. But is it possible in principle to define a nowhere dense ...
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140 views

Space of functions that are everywhere differentiable

Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
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For a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$, if $F$ is compact on compact subsets of $X$, then $F$ is compact on $X$

The problem as stated in the title isn't quite correct. Let $X$ be a topological space. What I have is a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$ which on compact subsets ...
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478 views

Pearson correlation and metric properties

Assuming that the data set was $z$-standardized to zero mean and unit variance (also assuming that it does not contain constant vectors). Then Pearson's r reduces to Covariance: $$\rho(X,Y) := ...
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40 views

How to prove this result about this space of sequences?

Let $s$ denote the metric space of all sequences of real or complex numbers with the following metric: $$ d( (\xi_j), (\eta_j) ) := \sum_{j=1}^{\infty} \frac{1}{2^j} \frac{|\xi_j - \eta_j|}{ 1 + ...
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How to prove the metric which defined by supremum of all semi-metric?

Define the function $f:X\times X \to R$ by $d(x,y)=\sup\{d_i(x,y):i\in I\}$, when each $d_i$ is a pseudometric; $d_i(x,y)=0$ need not imply $x=y$; for every $i$ in a directed set $(I,\leq)$ and $X$ is ...
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86 views

A question about a metric on $\mathbb{R}^\mathbb{N}$

Consider the metric space $(\mathbb{R}^{\mathbb{N}},d)$ where for $x,y\in\mathbb{R}^\mathbb{N}$ $$ d(x,y) = \sum_{n=1}^{\infty} 2^{- n} \frac{\bigvee_{k\leq n}\left|x_k-y_k\right|}{1 + \bigvee_{k\leq ...
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181 views

Showing a differential equation has a unique solution in $C[0, 1]$

Show that $$F(f)(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du$$ is a contraction on $(C[0, 1), d_u)$. Deduce that the differential equation $$(15 − 5t)\frac{df}{dt} = (5 + 3e^{t})f + ...
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245 views

Prove metric space…

Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$ If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by $$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$$ Prove that $d$ is ...
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Reference for the characterization of completeness for metric spaces

I have the following criteria for the completeness of a metric space that I want to use in some research paper. Let $(X,d)$ be a metric space. The following conditions are equivalent: (1) $X$ ...
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145 views

Convergence of a function in a metric space to its metric

Given a metric space $(\mathbb{A},d)$ with a metric $d$ being the Euclidean metric, if $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ is a matrix with the ...
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115 views

What is the correct distance measure for the (anti) de-Sitter space?

Given these two expressions 1) $\sinh{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1−(t^2−x^2)}}$ 2) $\sin{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1+(t^2−x^2)}}$ for distance $d$ from the origin $(0,0)$ to point $(x,t)$, ...
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Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
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151 views

What's the relationship between the riemannian metric and Jacobi field?

I encounter to the question in reading the following Excise: Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...
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82 views

Embedding tree metric isometrically into $\ell_\infty$

I just started (independent) learning on metric embeddings from the Fall 2003 offering of the course at CMU. I have a limited mathematical background and alas, it made me stumble at the first exercise ...
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223 views

Trouble with some equivalent conditions of compactness

I'm afraid this question may turn out to be a stupid one. Though it is related to a previous question of mine, I'll write it down in full. Let $(X, d)$ be a metric space (MS). I have to prove the ...
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108 views

Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$

I'm looking for examples of spaces $X$ such that: $X$ is a probability space. $X$ is a metric space. If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$. I ...
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29 views

Show that $ℓ_2(X)$ is Hilbert space for every set $X$

Show that $ℓ_2(X)$ is Hilbert space for every set $X$ I tryed to find a proof for this problem but i couldn't (searched on internet and mathematical books.Can we find a completed proof for this?
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38 views

What logical resources does one need to distinguish $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ from $\mathbb{Q}$?

(I'm inspired by this question.) As a strict lower bound, having just the order relation is not enough, since $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ and $\mathbb{Q}$ are both ...