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Questions tagged [metric-spaces]

A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.

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Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space

Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure. I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces $M^{s,p}(...
ABIM's user avatar
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Locally compact space that is not topologically complete

It is know that for a metric space, it is locally compact and separable iff exist an equivalent metric where a set is compact iff it is closed and limited. So, locally compact and seperable metric ...
Hugo Rafael Oliveira Ribeiro's user avatar
6 votes
1 answer
318 views

How complicated can the path component of a compact metric space be?

Let $X$ be a compact metric space and $P$ be a path component of $X$. Since we are not assuming $X$ is locally path connected, $P$ must need not be open nor closed. Certainly, $P$ must be separable ...
Jeremy Brazas's user avatar
6 votes
1 answer
284 views

Extending a partially defined metric on a metrizable space

Let $X$ be a metrizable topological space, $A\subseteq X\times X$ a nonempty closed subset which is reflexive, symmetric and transitive, $d:A\to \mathbb{R}_+$ a continuous function that satisfies the ...
omar's user avatar
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6 votes
1 answer
237 views

m-point-homogeneous, but not (m+1)-point-homogeneous

It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large). In other words, if $A\subset Q$ ...
Anton Petrunin's user avatar
6 votes
1 answer
298 views

A rather non-$F_\sigma$ Borel set

I asked this question at MSE a week ago, but received no answer, so I cross-post it here. I obtained a negative answer to this MSE question provided each metric space $X$ such that $|X|=\frak c$ and ...
Alex Ravsky's user avatar
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6 votes
1 answer
257 views

Expected doubling constant of a random Erdős–Rényi graph

Consider the $G(n,p)$ random graph model where $n$ is a ``large'' positive integer and $p\in (0,1)$. We may equip every realized random graph $G$ with its shortest path distance, making it into a (...
ABIM's user avatar
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When is a distance space dominated by a metric space?

A distance space is a pair $(X,d)$ where $X$ is a set and $d:X \times X \rightarrow \mathbb{R}$ is a symmetric, non-negative map such that $d(x,x)=0$ for all $x \in X$. These are sometimes called semi-...
David Bryant's user avatar
6 votes
0 answers
155 views

Metric spaces containing a topological disc

It is well-known that every connected, locally connected compact metrizable space $X$ contains an arc, that is, a subspace homeomorphic to $[0,1]$. Are there topological properties we can add to these ...
Jeremy Brazas's user avatar
6 votes
0 answers
111 views

A generalized Hausdorff dimension in form of a Lower semi continuous function

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
Ali Taghavi's user avatar
6 votes
0 answers
182 views

Factorization of metric space-valued maps through vector-valued Sobolev spaces

Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that $$ \int_{x\in X}\,d(y_0,f(x)...
ABIM's user avatar
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6 votes
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812 views

Limit of metric spaces

Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system. Assume ...
Giulio's user avatar
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What relates to measure spaces as topological spaces relate to metric spaces ?

Has there been study of a generalization of measure spaces along the following or similar lines ? Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball of radius $r$ of $U$ ...
user19172's user avatar
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5 votes
4 answers
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Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-...
Bazin's user avatar
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5 votes
3 answers
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Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube

Question: Is the hyperspace of the Hilbert cube $H=[0,1]^\mathbb {N}$ homeomorphic to $H$? Remarks and definitions: 1) The Hilbert cube $H$ is a compact metric space, where the metric is given by ...
Marcus's user avatar
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1 answer
415 views

Spreading $n$ points in $\{0,1\}^n$ as far as possible

Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$ We say that a positive integer $s$ is $...
Dominic van der Zypen's user avatar
5 votes
2 answers
2k views

Isometric embeddings of metric spaces in Hilbert spaces

There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: ...
Alex M.'s user avatar
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5 votes
1 answer
148 views

Continuity of central point operation

Stanisław Mazur and Stanisław Ulam, in their joint paper, characterized the mid-point $\ \frac{a+b}2\ $ in a Banach space in pure metric terms (without algebra). This allowed them to show that any two ...
Włodzimierz Holsztyński's user avatar
5 votes
1 answer
3k views

How is the notion of a Lipschitz structure on a manifold defined?

According to wikipedia, there is such a definition. $\:$ The candidate that I can come up with is "an equivalence class of metrics that induce the topology and make the space locally bi-Lipschitz to ...
user avatar
5 votes
1 answer
483 views

Can you always extend an isometry of a subset of a Hilbert Space to the whole space?

I remember that I read somewhere that the following theorem is true: Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
Cosine's user avatar
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1 answer
200 views

Criterion for Kuratowski Limit Inferior

Let $(X,d_X)$ be a compact metric space and let $\{K_n\}_{n=1}^{\infty}$ be a collection of non-empty compact subsets. Let $K\subseteq X$ be compact. Then, if for every $x_n \in K_n$ we have $$ d_X(...
SetValued_Michael's user avatar
5 votes
1 answer
982 views

sets without perfect subset in a non-separable completely metrizable space

Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements? [1] $D$...
Arkadi's user avatar
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5 votes
2 answers
642 views

Is the Hausdorff metric on sub-$\sigma$-fields separable?

Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ ...
Jason Rute's user avatar
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5 votes
1 answer
390 views

Topological properties inherited by the Hausdorff metric space

Let $(X,d)$ be a metric space and $(K_X , h_d)$ be the associated metric space of nonempty compact subsets of $X$ with the Hausdorff metric. It is well known that $K_X$ inherits certain topological (...
Logan Fox's user avatar
  • 267
5 votes
2 answers
448 views

Space of curves

I am reading Burago, Burago & Ivanov's book where they distinguish the notion of a curve and a path in the following way: a path in a topological space $X$ is simply a (continuous) map from a ...
erz's user avatar
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5 votes
2 answers
2k views

Distance between two metric spaces

I am given two metric spaces as two arrays of the same size. Each one is supposed to represent distance between vertices on a mesh in R^3. The meshes are assumed to have the same number of vertices ...
Steve's user avatar
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1 answer
168 views

Compactness of symmetric power of a compact space

Suppose I have a compact metric space $(X,d)$ and let $\mathcal{X}=X^K$ be the product space. Consider the equivalence relation $\sim$ on $\mathcal{X}$ given as: for $\alpha,\beta\in \mathcal{X}$, $\...
Sunrit's user avatar
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5 votes
1 answer
337 views

Greedy simplices in an ultrametric space (generalized Bhargava $p$-orderings)

Let $\left(U, d\right)$ be a finite ultrametric space -- that is, $U$ is a finite set, and $d : U \times U \to \mathbb{R}_{\geq 0}$ is a metric on $U$ such that every $x, y, z \in U$ satisfy $d\left(x,...
darij grinberg's user avatar
5 votes
1 answer
198 views

Iterating the dimensional kernel of a metric space

Fix $n\in \mathbb N$. Let $X$ be a separable metric space of (inductive) dimension $n$. Let \begin{align} \Lambda(X)&=\{x\in X:X\text{ is $n$-dimensional at }x\}\\ \\ \Lambda^2(X)&=\Lambda(\...
D.S. Lipham's user avatar
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5 votes
0 answers
158 views

Does "achieving more GH-distances than some compact space" imply compactness?

Previously asked and bountied at MSE: For complete metric spaces $X,Y$, write $X\trianglelefteq Y$ iff for every complete metric space $Z$ such that the Gromov-Hausdorff distance between $X$ and $Z$ ...
Noah Schweber's user avatar
5 votes
1 answer
129 views

Are normal metric currents dense in the space of all metric currents?

It is classical that Euclidean normal currents are dense in the space of all currents. This can be achieved through mollification. What I want to know if this is still true for metric currents. In ...
Lolman's user avatar
  • 391
5 votes
1 answer
373 views

Points of differentiability of squared distance from a point in metric spaces

I posted this same question on MSE with no answer. Let $I:=(0, + \infty)$ and let $(X,d)$ be a complete and separable metric space. In this setting we say that $u : I \to X$ is absolutely continuous ...
Bremen000's user avatar
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5 votes
0 answers
296 views

For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?

Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $\phi \colon (M_1, d_1) \to (M_2, d_2)$ is a large-scale Lipschitz essentially surjective map if there exist constants $A \geq 1, B \geq 0$,...
Carl-Fredrik Nyberg Brodda's user avatar
5 votes
0 answers
205 views

A generalization of SOCA

Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...
Iván Ongay Valverde's user avatar
4 votes
2 answers
440 views

largest diameter of intersection of two balls

Two closed balls with a common radius are positioned so that the centre of either ball is on the boundary of the other. I am interested in the extremal diameter of their intersection, in an arbitrary ...
András Salamon's user avatar
4 votes
2 answers
399 views

Terminology for metrics?

For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for ...
Miel Sharf's user avatar
4 votes
2 answers
279 views

Hausdorff dimension of sequence space

Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x_n),(y_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,...
Ian Short's user avatar
4 votes
2 answers
309 views

Finitely isometrically persistent metric spaces

The goal of this question is to develop further the discussion initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was ...
Mikhail Ostrovskii's user avatar
4 votes
2 answers
191 views

Reference request: "Tangent relation" in metric spaces

Let $X,Y$ be metric spaces. Let $f,g : X \to Y$ be two maps and $x_0 \in X$. Let us say that $f$ and $g$ are tangent at $x_0$ if the following condition is satisfied: For every $\epsilon > 0$ there ...
Martin Brandenburg's user avatar
4 votes
2 answers
374 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
Analyst's user avatar
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4 votes
1 answer
292 views

Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry?

I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an ...
Saúl RM's user avatar
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4 votes
1 answer
279 views

Product topology from two premetric spaces induced by sum of premetrics?

For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$. Do ...
fsp-b's user avatar
  • 463
4 votes
1 answer
273 views

Relation between two permutation metrics

Note: I asked this question a few months ago here, but received no answer. Consider the following two metrics on permutations of $\{1,2,\dots,n\}$: $d_\text{swap}(\sigma,\tau)$ is the minimum number ...
reservoir's user avatar
4 votes
1 answer
183 views

Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same

I'm reading Heinonen's book on metric measure spaces. He writes that for general domains $\Omega \subset \mathbb{R}^n$, $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$ where the former are Hajlasz-Sobolev ...
yoshi's user avatar
  • 427
4 votes
1 answer
210 views

Bi-Lipschitz embeddings of compact doubling spaces

Suppose that $(X,\rho)$ is a compact doubling metric space. Does there necessarily exist an $\epsilon>0$ and a maximal $\epsilon$-net $\{x_i\}_{i=1}^n\subseteq X$ such that the map $$ \begin{...
ABIM's user avatar
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4 votes
1 answer
159 views

Extending a metric in a bi-Lipschitz way

Suppose we are in the following situation: $(X,d)$ is a metric space and $Y$ is a subspace of $X$. Furthermore we have a different metric $\delta$ defined on $Y$ such that $\delta$ is bi Lipschitz ...
an_ordinary_mathematician's user avatar
4 votes
1 answer
407 views

Lipschitz-regularity of partition of unity

Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
Carlos_Petterson's user avatar
4 votes
1 answer
370 views

Inducing metric spaces

Let $f\colon \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$ be a function. We say that $f$ has the property of inducing metric spaces, whenever for all metric space $(X,d)$, $(X, f \circ d)$ is also a ...
calc's user avatar
  • 283
4 votes
1 answer
97 views

Inner regularity property of covering number of metric spaces

Let $(X,d)$ be a complete metric space and $n\in\mathbb N$. Suppose that every finite subset $F\subset X$ can be covered by $n$ closed balls of $X$ (that is, $N(Y,d,1)\le n$, in terms of covering ...
Pietro Majer's user avatar
  • 60.5k
4 votes
1 answer
104 views

Generalization of a bounded variation

Let $(X, d)$ be a metric space. We will say that $\gamma \colon [a,b] \to X$ is of bounded variation, if \begin{equation} V(\gamma) = \sup_{a=t_0 < \cdots < t_n < b} \sum_{i=1}^n d( \gamma(...
Kacper Kurowski's user avatar

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