Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then $\...
22
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1answer
742 views

Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?

Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb R^{n-...
21
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1answer
300 views

a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
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$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^...
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445 views

What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
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388 views

When is an Open Set Homeomorphic to the Interior of its Closure?

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$. I am looking for known assumptions on $X$ and $U$ such that one of the following ...
11
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243 views

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, $\left(-\frac{1}{2},0\...
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156 views

Contracting subsets

Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
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161 views

Is the barycenter of a convex curve in $\mathbb R^2$ Lipschitz with respect to the Hausdorff distance?

For a curve $C$, its barycenter is $$\text{Bar}(C) = \frac{1}{\text{length}(C)}\int\limits_C x d \mathcal H^1(x).$$ Does there exist a constant $L$ such that for $C_1,C_2$ convex curves in the plane, $...
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283 views

Uniform Spaces: Completeness

Attention This thread has been generalized to uniform spaces as general metric spaces. Context The context was the equivalence: $$K\text{ compact}\iff K\text{ totally bounded, complete}$$ That is a ...
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601 views

Non empty set with zero diameter

Let $A \subset X$ where $X$ is a metric space. by definition diam$(A) = \sup\{ d(x,y), x,y \in A\}$. if $A$ is non empty and has zero diameter, can I conclude that $A$ is a singleton? i reason as ...
9
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337 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ It'...
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134 views

What is the epistemological status of the usual proof(s) of Pythagoras' theorem?

Pythagoras' theorem has a variety of geometric proofs, such as: I want to teach at least one of these proofs to my high school students, because it shows that the formula $\|(x,y)\| = \sqrt{x^2 + y^2}...
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184 views

A generalization of the abc-conjecture?

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$. Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$. Using this one can define similarities over the natural numbers, ...
8
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363 views

A definition of differentiable functions for arbitrary topological spaces

Background It is well-known that there is no notion of derivative for arbitrary topological spaces. However while investigating the notion of derivative as we find in one variable real analysis I came ...
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243 views

Hyperbolic metric spaces 2

I am trying to prove a lemma in Burago's "A Course in Metric Spaces" (Exercise 8.4.4, p.286). Here is a link to a different person's question about the very next exercise in that book, which also ...
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Proving $\mathbb{R}$ is Hausdorff with final topology induced by a function $f$.

Consider a Hausdorff topological space $(X,\tau)$. Suppose $(X,\tilde{\tau})$ is the minimal normalization of $(X,{\tau})$, that is, for every given normal topology $\sigma$, where $\tau \subset \...
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2answers
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The Distance between two points in a hypothetical universe.

I have a hypothetical universe where the distance between two points in spacetime is defined as:$$ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2+dz^2$$Where $\phi$ has units of $km\space s^{-2}$. The space in ...
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In which $L^p$ metric is $\pi = 3.5$?

In which $L^p$ metric is $\pi = 3.5$? I am interested because it's well known that $\pi$ can range from $3.14...$ to $4$ in $L^{\infty}$
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If $E$ is a totally bounded subset of a metric space $X$. Then, any subset of $E$ is totally bounded.

Proof: Let $D \subset E$, where $E \subset X$ and ($X,d$) is a metric space. Suppose that $E$ is totally bounded. That is: for all $\varepsilon > 0$, there exist finitely many points $x_1, \ldots , ...
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$W^{1,p}$ is separable for $1\leq p<\infty$

I've been asked to prove that the Sobolev spaces $W^{1,p}(\Omega)$, $\Omega$ open in $\mathbb R^n$, are separable for $1\leq p <\infty$ using the map $$i\colon W^{1,p}(\Omega)\to L^p(\Omega)\times ...
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600 views

What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
7
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2answers
158 views

A basic question on symmetry of metric space

In the metric space definiton, the second condition for a metric i.e. symmetry (d(p,q)=d(q,p)) is present. But, I have not seen any example where this condition has been used. Can anyone give any such ...
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910 views

Topological necessary and sufficient condition for tightness

Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$: For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
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Taxicab metric *with stoplights*; does it ever give the Euclidean metric?

This questions is not terribly formal in nature, but please bear with me: what I'm looking for is a model of a traveling via taxi on a grid of streets where the "metric" in some sense ...
6
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57 views

How to show that $\left\|F_n^* - F\right\|={{O}_{p}}\left({{n}^{-\frac{1}{2}}}\right)$

Let $X_1,\ldots,X_n$ be iid from a cdf $F$ on $R^d$, $X_1^*,\ldots,X_n^*$ iid from empirical cdf $F_n$. Let $F_n^*$ be empirical cdf based on $X_i^*$'s. Using DKW inequality, Let $\rho\left(F_1,F_2\...
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203 views

Questions about pseudo metric on quotient space

Let $(X,d)$ be a metric space and $\sim$ be an equivalence relation on $X$, then we can form the quotient space $X/\sim$. We can also define a pseudo metric on the set of equivalence classes as ...
6
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89 views

Checking whether $X=\mathbb R$ with $d(x,y)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ is a metric space

Examine whether $d$ is a metric on $X=\mathbb{R}$ where $d\left(x,y\right)=\min\{ \sqrt{|x-y|},|x-y|^2\}$ for all $x,y\in \mathbb{R}$ I think it is not. Even though it satisfies all other properties ...
6
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1answer
739 views

Consider the metric space of infinite sequences of 0s and 1s under this metric.

For $x, y ∈ \{0, 1\}^\mathbb{N}$, define $d(x, y) = 2^{-n}$ where $n$ is the first position where the sequences $x$ and $y$ are different. Show that ($\{0,1\}^\mathbb{N}, d$) is compact. Show that ...
6
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188 views

Norm topology vs metric topology

I have been studying functional analysis and the notion of a Frechet metric frequently comes up. Not every Frechet metric induces a norm topology, and it is said that this results in a more ...
6
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Is the Hausdorff distance between $I=[0,1]$ and $B_n = \{0, \frac{1}{n}, \cdots, \frac{n-1}{n}, 1\}$ $0$ or $\frac{1}{2n}$?

In this post, it is claimed that the Hausdorff distance $d_H$ between $I=[0,1]$ and $B_n = \{0, \frac{1}{n}, \cdots, \frac{n-1}{n}, 1\}$ equals zero. Is it not equal to $\frac{1}{2n}$ ? For ...
6
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295 views

Find distance to a set (subspace) without computing closest point

General setup: we have a finite-dimensional normed linear space $(V, \| \cdot \|)$, a subspace $U \subset V$, and a fixed vector $v_0 \in V$. We want to find the distance between $v_0$ and $U$. (No ...
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375 views

Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
6
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676 views

Equivalent definition of Cauchy sequence

A sequence $x_i$ is Cauchy if for all $r>0$, there exists $n$ s.t. $i,j\geq n$ implies $d(x_i,x_j)<r$. My question is, is it equivalent to define Cauchy as follows? $x_i$ is Cauchy if for all $...
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Metric on total space, connection and Hopf fibration

As is well known that the three sphere $S^3$ may be viewed as a $S^1$ bundle over base space $S^2$. And it is more interesting, but likewise confusing to me that the metric on $S^3$ may be written as: ...
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Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in V_\...
6
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852 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha z_{\bar{\beta}}}...
6
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Selecting a subset from a set such that a given quantity is minimized

Let $A$ be is a set of some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^A$ denote the mean Euclidean distance from the point $x$ to its $k$ nearest points in $A$ (others than $x$). Let $C \...
6
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0answers
1k views

Completeness of a metric space with the Hausdorff metric

Let $(Y,d)$ be a metric space and let $K(Y)$ denote the set of all non-empty compact subsets of $Y$. This collection is a metric space when equipped with the Hausdorff distance $h$. I want to prove ...
6
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277 views

Examples of Moscow spaces

A space $X$ is called Moscow, if for each open subset $U$ of $X$, the closure of $U$ in $X$ is the union of a family of $G‎_{δ}$‎‎‎-subsets of $X$ . For example, every first countable $T_1$-space is ...
6
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638 views

Are these sets in $\mathbb{R}$ open and/or closed: $\{\frac{1}{n} : n \in \mathbb{N}\}$, $\{0\}\cup \{\frac{1}{n} : n \in \mathbb{N}\}$ and $[0,1)$.

In $\mathbb{R}$, are these sets open? Are they closed? $A = \{\frac{1}{n} : n \in \mathbb{N}\}$ $B = A \cup \{0\} $ $[0, 1)$ My thoughts: $A$ is not open as if we have an open ball with $r > 0$ ...
6
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235 views

Virtually cyclic groups

Let $G$ be a group with finite generating set $A$, define the distance $$d(g,h)=\mathrm{min}\{n:gh^{-1}=a_1^{\varepsilon_1}\dots a_n^{\varepsilon_n}, a_i\in A,\varepsilon_i=\pm1\}.$$ Define the ball ...
5
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53 views

question about group actions on metric spaces

Suppose we have a metric space $V$, a group $G$ and an action $\cdot: G \times V \rightarrow V$. What assumptions must I make so that the following is true? Claim: For each $x, y \in V$, if there ...
5
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0answers
146 views

Why aren't derivatives defined on metric spaces?

I'm studying the book by Ambrosio on gradient flows in metric spaces. It's stated that the usual notion of gradient flow, $$ x'(t) = -\nabla_x f(x),$$ is not defined on metric spaces because we don't ...
5
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1answer
102 views

If $(X, \|\cdot\|)$ is a normed vector space, then $(X\setminus\{0\}, d)$ is a metric space for $d(x, y) = \frac{\|x-y\|}{\|x\|+\|y\|}$

Consider the following statement: If $(X, \|\cdot\|)$ is a normed vector space, then $(X\setminus\{0\}, d)$ is a metric space for $$d(x, y) = \frac{\|x-y\|}{\|x\|+\|y\|}$$ I'm trying to figure out ...
5
votes
0answers
103 views

A rather non-$F_\sigma$ Borel set

I obtained a negative answer to this question provided each metric space $X$ such that $|X|=\frak c$ and density $d(X)<\frak c$, contains a Borel set $B$ such that $|B\setminus C|=\frak c$ for each ...
5
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0answers
82 views

Prove that there exists $r>0$ such that the set $\{y \in \Bbb R^n\ :\ \|y - x\| \le r\ \text {for some}\ x \in K\}$ is a compact subset of $\Omega.$

Let $\Omega \subseteq \Bbb R^n$ be an open set and $K \subseteq \Omega$ compact. Prove that there exists an $r \gt 0$ such that the set $$\left \{y \in \Bbb R^n\ :\ \|y-x\| \leq r\ \text {for some}\ x ...
5
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94 views

Embedding whole flat 2-Torus in $\Bbb R^3$

I am trying to better understand an isometric embedding of a flat 2-Torus in $\Bbb R^3$ via a $C^1$ map. Here is a visualization involving $C^1$ fractals: This embedding is warped by an infinite ...

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