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) $d(x,y)+d(y,z) \geq d(x,z)$.
442 questions
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Continuous inclusion of metric spaces of smaller capacity
If $(X,d_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d_X)$, that is
$$
\kappa_X(\epsilon)\leq \...
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Spreading $n$ points in $\{0,1\}^n$ as far as possible
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
We say that a positive integer $s$ is $...
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Stretching map of $n$ points from $\{0,1\}^n$ to $\{0,1\}^{n+1}$ with respect to their Hamming distance
Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
Given an integer $n>0$ and a set $S\...
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Is every finite graph isomorphic to the proximity graph of some $S\subseteq \mathbb{R}^n$?
This is the question that I should have asked before asking this older question.
If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ ...
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Spatial dimension of a finite graph
If $(X,d)$ is a metric space, we associate with it a simple, undirected graph, called its proximity graph $G(X,d)$ given by $V(G(X,d)) = X$ and $$E(G(X,d)) = \big\{\{x,y\}:x\neq y\in X \text{ and } d(...
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Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces
Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...
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An extension for lower semi continuous lower bounded real valued functions class
Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...
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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 ...
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Graphs represented by a subset of a metric space
Let $(X,d)$ be a metric space, and suppose $S\subseteq X$ is a finite subset in which all pairwise distances are distinct (formal definition here).
If $x\in S$ and $k$ is a non-negative integer with $...
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Weak convexity in graphs
I asked the following question in MathStackExchange, but I did not get any answer, and I think that this might be the appropriate venue for the question.
As we know, a finite undirected graph ...
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Extending homeomorphisms between compact metric subsets
Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism.
Is it possible to extend $h$ to a ...
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Intersection of nested open ball in complete metric spaces is nonempty?
My question is that whether the following statement is true or not.
In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies
$$
\exists \epsilon > ...
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Measure on group invariant under group action on metric space
This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO.
The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...
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A property on some unbounded metric spaces
Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property:
$\forall x, \forall y \in X, \exists \{z_n\}$ such that $\lim_{n\to +\infty } d(x,z_n)=+\infty$ and $\...
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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: ...
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Quotient of compact metrizable space in Hausdorff space
Lets $X$ be a compact metrizable space and $f:X\to Y$ be a quotient map such that $Y$ equipped with the quotient topology is Hausdorff. Thus $Y$ is metrizable. Lets $\sim$ be an equivalence relation ...
CommunityBot
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Balls in Lawvere metric spaces
Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no ...
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Universally meager spaces and large cardinals
Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
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Path Metric Topology
Is there an example of a metric space $(X,d)$ whose corresponding path metric, $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$?
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On the Large Cardinal Strength of Normal Moore Space Conjecture
In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces:
Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable.
Then ...
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Are there any statistical metrics that satisfy this kind of condition?
Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f_1 = N(\mu+\epsilon,\sigma^2)$ and $f_2=N(\mu,(\sigma+\epsilon)^2)$ be some small ...
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Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?
I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions
It has a ...
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Li-Yorke chaos: the non compact case
1) Is there any notion of Li-Yorke chaos for non compact (metric) spaces $X$ and non continuous transformation $f:X \rightarrow X$? Could you bring some references?
2) I mean, why are so important ...
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Totally bounded spaces and axiom of choice
Wikipedia article on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it ...
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How to show the cardinality of nonisometric compact metric spaces is the continuum
It is asserted in A Course in Metric Geometry by Burago, Burago, Ivanov that
there can be no more than continuum of mutually nonisometric compact spaces
How is this proven?
Its clear that there ...
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Does rate of convergence in probability come from a metric?
In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
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Base of topology for metric-like space
Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)...
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Hausdorff convergence of preimages of discrete-valued functions
Suppose $f_n$, $f:X\to K$ where $K$ is a finite set and $(X,d)$ is a metric space. Suppose also that $f_n(x)\to f(x)$ for all $x\in X$ (pointwise convergence). Finally, let $d_H$ be the Hausdorff ...
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Gromov-Hausdorff relative compactness without curvature restrictions
A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by ...
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Is there a "universal" connected compact metric space?
Fact 1. The Cantor set $K$ is "universal" among nonempty compact metric spaces in the following sense: given any nonempty compact metric space $X$, there exists a continuous surjection $f\colon K \to ...
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Graph contained in a metric space
I have a metric space $X$ and a graph $G=(V,E)$ whose set of vertices is a subset $V\subset X$ (and $E$ is the set of edges, which is a symmetric subset of $V\times V$). For each $v\in V$, the set of ...
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A Riemannian metric on the plane such that the intersection of every two discs is a disc, again
Is there a Riemannian metric on $\mathbb{R}^2$ (or a $2$ dimensional manifold) such that the intersection of every two open discs is an open disc, again?
As linear version of this question we ask:
...
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$\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space
Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:
$X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
$X' \cong \ell_\infty ...
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A characterization of Cauchy filters on countable metric spaces?
Given a filter $\mathcal F$ on a countable set $X$, consider the family
$$\mathcal F^+:=\{A\subset X:\forall F\in\mathcal F\;(A\cap F\ne\emptyset)\}.$$
The following characterization is well-known.
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Existence and uniqueness of fixed point in generalized condition of triangular norm
Definition 1) A Menger space is defined as a triple $\left( S,F,T \right)$ where $S$ is a set , $F$ is a collection of distribution functions and $T$ is a triangular norm function
$T:[ 0,1 ]\...
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Given a topological field with non-trivial topology given by a metric, is the metric necessarily translation-invariant? [closed]
If the topology is locally compact and non-trivial, then the metric is usually translation-invariant or even arises from a norm. But is it necessary to be invariant? I failed to give a counterexample.
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Ultraproduct of metric spaces
Let $I$ be a set and $\mathcal{U}$ be an ultrafilter on $I$. Suppose that $(X_{i}, d_{i})_{i\in I}$ is a family of pointed metric spaces with a distinguished point $e_{i}$ for each $i\in I$. We set
$$...
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Approximation of semicontinuous function
I'm looking for a reference for the theorem saying that a real-valued lower (upper) semicontinuous function on any metric space can be reached as a pointwise limit by a non-decreasing (non-increasing) ...
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In a metrizable order topology, is the order relation compatible with the metric? [closed]
Does $x \le y \le z$ imply $d(x, y) \le d(x, z)$?!
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Manhattan distance vs. absorption time on an unbounded integer lattice
Imagine I have unbounded $d$-dimensional integer lattice where I take two vertices, $v_a$ and $v_b$, separated by a fixed Manhattan distance $L$, and I release a random walker at $v_a$ and allow for ...
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Results that are easier in a metric space
Are there any significant results in the theory of metric spaces that (are considerably more difficult to reproduce/have not been reproduced) in the theory of uniform spaces?
In particular, I'm ...
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When is "metric dimension" well defined?
A subset $B$ of a metric space $(M,d)$ is called a metric generating set if and only if $$[\forall b \in B, d(x,b)=d(y,b)] \implies x = y \,. $$ A metric generating set $B$ is called a metric basis ...
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Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
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Cauchy subsequences in "Hausdorff Cauchy sets"
This is a follow-up to an older question.
Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of bounded non-empty subsets of $X$ such that for all $\varepsilon &...
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Hausdorff distance and Cauchy sequences
This is a generalization of an older question.
Let $(X,d)$ be a metric space and let $(A_n)_{n\in\mathbb{N}}\subseteq X$ be a sequence of non-empty closed subsets such that for all $\varepsilon > ...
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Lipschitz extensions preserving the convex hull of the range
We assume that $X$ is a metric space and that $A \subseteq X$ is a subset. Let $f : A \rightarrow \mathbb R$ be a Lipschitz-continuous function with Lipschitz constant $L$.
By the Kirszbraun theorem, ...
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How to prove that $k(x)$ is not complete in the $x$-adic metric [closed]
It is not hard to find proofs showing that $\mathbb{Q}$ is not complete with respect to the metric induced by the valuation $|\;\;|_p$.
For example, it is enough to recall that every complete metric ...
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Sources for Alexandrov surfaces
There are two distinct notions in differential geometry associated
with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded
from below; (2) Alexandrov surfaces of bounded total curvature (...
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Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
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Generalize upper semicontinuous regularization using Borel Hierachy
Let $X$ be a metric space. Suppose a real-valued function $f:X\rightarrow \mathbb{R}$ is upper semicontinuous class $2$ if for all $c \in \mathbb{R},$ its preimage $f^{-1}(-\infty,c)$ is $F_{\sigma}$.
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