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) $d(x,y)+d(y,z) \geq d(x,z)$.
476 questions
2 votes
0 answers
69 views
Adjoint functor for thickening covers in metric spaces
Let $\mathbf{MetP}$ denote the metric space pairs $(X,E)$ with $X\subset E$, morphism $(X,E)\to (X',E')$ is an 1-Lipschitz map $f$ s.t. $f(X)\subset X'$. Let $\mathbf{MetPC}$ denote the pairs $(X, E, \...
- 121
3 votes
1 answer
490 views
Example of connected, locally connected metric space that isn't path-connected?
I tried to construct a reasonable example for someone, meshing together various sorts of dust, but I either failed or wound up with sets whose separation properties are far too delicate for the ...
- 559
0 votes
0 answers
84 views
Embeddings sets of functions into Lipschitz functions
Let $X$ be a set, $\mathcal{M}(X)$ denote the set of metrics on $X$, and fix a $\mathcal{F}\subseteq \mathbb{R}^{\mathcal{X}}$. Let $L \in\mathbb{R},\delta\ge 0$. For any metric $\rho$ on $X$, we ...
- 4,948
1 vote
0 answers
232 views
Definition of length structure in Burago, Burago, Ivanov's "A Course in Metric Geometry" does not imply Exercise 2.1.4?
In Burago, Burago, Ivanov's "A Course in Metric Geometry" (Definition 2.1.1, page 26 and 27) a length structure on a topological space $X$ is defined as a pair $(A,L)$ where $A$ is a set of ...
- 684
6 votes
0 answers
229 views
Reference request: Katetov's 1988 paper "On universal metric spaces"
I am reading Itaï Ben Yaacov's 2014 paper "The linear isometry group of the Gurarij space is universal", and it references a paper of Katětov's called "On universal metric spaces" ...
- 61
1 vote
0 answers
130 views
Completion of a quasi-normed space
Let $X$ be a quasi-normed space, which is not necessarily complete. This is a metric (with respect to the equivalent $p$-norm) space. So, we can consider the completion $\bar{X}$. However, for $x\in \...
- 685
8 votes
0 answers
201 views
A natural metric on commutative rings
Let $R$ be a commutative ring. For an element $r\in R$, let $|r|$ denote the nilpotency degree of $r$. (If $r$ isn't nilpotent, we define $|r|=\infty$.) For a pair of elements $x,y\in R$, define: $$...
- 2,171
5 votes
1 answer
252 views
A property between zero-dimensional and strongly zero-dimensional for metric spaces
I was recently reading Ordnungsfähigkeit total-diskontinuierlicher Räume by Herrlich which shows that a strongly zero-dimensional metrizable space is a LOTS. I've noticed that in the article they go ...
- 2,307
2 votes
0 answers
125 views
Has there been any study of normed semilattices?
First of all, I should confess that this is not my field of study. Recently, I encountered a situation where I had a (meet) semilattice $\mathcal{L}$ and a compatible valuation (positive semidefinite ...
- 1,203
4 votes
1 answer
310 views
Is the compact-open topology Fréchet–Urysohn?
Let $(X, d)$ be a complete and separable metric space. Let $\mathcal C (X)$ be the space of all continuous real-valued functions on $X$. We endow $\mathcal C (X)$ with the topology induced by uniform ...
- 1,163
3 votes
0 answers
145 views
Spaces of Lipschitz functions
$\DeclareMathOperator\Lip{Lip}X$ and $Y$ are compact metric spaces. $\Lip((X,d), (Y,\rho))$ is all Lipschitz maps from $X$ to $Y$. Is there a topology on $\Lip((X,d), (Y,\rho))$ that makes it a Baire ...
9 votes
1 answer
261 views
Does every metric space of the form $X^2$ embed isometrically into some finitely iterated hyperspace of $X$?
This question occurred to me while I was writing my most recent answer. Recall that given an extended metric space $(X,d)$ the hyperspace of $X$, which I'll write as $\def\Hc{\mathcal{H}}\Hc(X)$, is ...
- 19.2k
5 votes
0 answers
136 views
Find a fixed-point-free continuous map for a topological space $X\times X$ where $X$ has the fixed-point property
A topological space $X$ has the fixed-point property (FPP) if, for every continuous map $f:X\to X$, there is an element $x\in X$ such that $f(x) = x$. The following is an example of a space $X$ with ...
- 1,903
1 vote
0 answers
42 views
Covering Number of Tensorized Class
Let $\mathcal{F}_1,\mathcal{F}_2$ be sets of continuous functions from $[0,1]$ to $[0,\infty)$ and suppose that, for every $\varepsilon>0$, then $\varepsilon$-covering numbers in the uniform norm ...
2 votes
1 answer
273 views
Is any arc in a metric space "essentially" expanding?
Let $(X,d)$ be a geodesic metric space (it is all right to assume $X$ is compact), let $\gamma:[0,1]\to X$ be an injective, continuous map. Is there a geodesic metric $d_2$ on $X$ which is equivalent ...
- 13k
13 votes
1 answer
453 views
Does the statement that every sequentially compact metric space is compact imply the axiom of countable choice?
I know that the statement that every sequentially compact pseudometric space is compact implies the axiom of countable choice (H. Herrlich, Axiom of Choice, 2006). How about in the realm of metric ...
- 447
3 votes
1 answer
234 views
Cantor sets without small clopen balls
Every topological copy $C$ of the Cantor set inside $\mathbb R$ (with the inherited euclidean metric) has arbitrarily small clopen balls centered at arbitrary point. The same is true if $C$ is ...
- 1,100
4 votes
1 answer
208 views
Non-existence of any cluster point of the sequence $(y_j)$ in a cofinally Bourbaki complete metric space
I am studying the paper, New Types of Completeness in Metric Spaces (Annales Academiæ Scientiarum Fennicæ Mathematica, Vol. 39, 2014, pp. 733–758 doi:10.5186/aasfm.2014.3934). I am currently working ...
- 43
11 votes
1 answer
293 views
Is every metric on $[0,1]^n$ satisfying the midpoint property induced by a norm?
Let $\Omega\subseteq\mathbb{R}^n$ be a bounded open convex set. A metric $d:\Omega\times\Omega\rightarrow\mathbb{R}^n$ satisfies the midpoint property if for any $x,y\in\Omega$, we have $$d(x,m) = d(y,...
- 113
7 votes
1 answer
399 views
When do the "Gaifman metrics" not vary much?
Given a structure $\mathfrak{A}=(A;...)$ in a finite language $\Sigma$, let $\Phi_\mathfrak{A}$ be the set of all finite tuples of $\Sigma$-formulas $\overline{\varphi}$ such that $\mathfrak{A}_{\...
- 19.3k
0 votes
0 answers
155 views
Homeomorphism of tangent spaces of Alexandrov spaces
Let $f\colon X\to Y$ be a homeomorphism of finite dimensional Alexandrov spaces with curvature bounded below. Question: Is it true that for any point $p\in X$ the tangent spaces $T_pX$ and $T_{f(p)}Y$ ...
- 23k
4 votes
1 answer
275 views
Is metric entropy characterized by dense subsets?
Let $X\subseteq Y$ be a dense subset of a metric space $(Y,\rho)$. Let $\varepsilon>0$, $A\subseteq Y$ and let $N(A,\varepsilon)$ denote the external $\varepsilon$-covering number of $A$; i.e. the ...
0 votes
0 answers
38 views
Are there known structural obstructions or theorems about partial-distance Katětov expansions that might fail to encode a TSP for large instances?
I have been experimenting with a partial-distance encoding of the decision version of the Traveling Salesman Problem (TSP) using a combination of Katětov–Urysohn ideas and what I have been calling “...
- 1
1 vote
0 answers
215 views
Approximate metric midpoints and uniform classification of Banach spaces
The approximate midpoint method is a classical technique in nonlinear theory that was developed by Enflo to show that $\ell_{1}$ and $L_{1}$ are not uniformly homeomorphic. Let $(M,d)$ be a metric ...
- 51
0 votes
0 answers
131 views
Moebius bands collapsing to segment
Let $M$ be a (compact) Moebius band. Let $\{g_i\}$ be a sequence of Riemannian metrics on $M$ with uniformly bounded below Gauss curvature and such that the boundary of $M$ is geodesically convex. Is ...
- 23k
4 votes
0 answers
226 views
Does every compact metric space admit a finite contracting family of maps?
Let $X$ be a metric space. A collection $\mathcal F$ of maps $X\to X$ (not necessarily continuous) is called contracting if $$\forall \varepsilon>0\ \exists n\in\mathbb N\ \forall f_1,\dots,f_n\in\...
- 1,100
3 votes
0 answers
97 views
Finite-Dvoretzky theorem variant with codomain of prespecified dimension
Let $n,k\in \mathbb{N}_+$ and $F_k\subset \ell^n_2$ be a $k$-points subset of $\ell_n^2:=(\mathbb{R}^n,\|\cdot\|_2)$. How well can $F_k$ be bi-Lipschitzly embedded into $\ell^N_{\infty}$? ...
- 4,948
0 votes
0 answers
125 views
Smallest enclosing balls
Let $(M, d)$ be a metric space. For a finite subset $\sigma$, we define $r(\sigma)$ to be inf$\{s: \exists y\in M \text{ such that }\sigma\subseteq B[y, s]\}$, here $B[y, s]=\{x\in M: d(y, x)\leq s\}$....
1 vote
0 answers
146 views
A geometrically-constrained Wasserstein distance
Let $X$ be a non-empty finite set, let $\mathcal{M}(X)$ denote the set of metrics on $X$ for which if $\rho\in \mathcal{M}(X)$ then $\rho(x,y)\ge 1$ for all $x,y\in X,\, x\neq y$ (i.e. distinct points ...
- 4,948
2 votes
1 answer
373 views
Given a metric space $M$ and $a,b \in M$, does there exist a compatible metric $d$ under which $d(a,c)+d(c,b) = d(a,b)$ for all $c \in M$?
Every separable metric space $M$ embeds homeomorphically in the Hilbert cube $H = [0,1]^\omega$. Since the cube is $2$-homogeneous (indeed $n$-homogeneous for any $n$) we can assume any two given ...
- 2,085
0 votes
0 answers
89 views
Terminology: What do I call a local equivalence class of metrics?
Let $X$ be a locally compact topological space. On each compact subset $K$ of $X$ I have a Lipschitz-equivalence class of metrics on $K$, call this equivalence class $M_K$, satisfying the obvious ...
- 323
3 votes
0 answers
78 views
Reference request for Cauchy completeness of generalized metric spaces
I am looking for a reference to where basic properties of Cauchy completeness are developed for generalized metric spaces whose distance is in a commutative linear quantale (or equivalently a complete ...
- 2,301
4 votes
0 answers
155 views
Spaces Admitting Geodesic "Tilings"
Let $(X,d)$ be a uniquely geodesic intrinsic metric space, for any $N\in \mathbb{N}_+$ let $\Delta_N:=\{w\in [0,1]^N: \,\sum_{n=1}^N\, w_n=1\}$. Background: Let $\eta:\Delta_N\times X^n \to X$ be a ...
- 4,948
1 vote
0 answers
123 views
Does a quadratic form depend locally Lipschitz-continuously on the related ellipse?
Let $Q_+$ be the space of positive definite quadratic forms on $\mathbb R^2$, equipped with the metric arising from thinking of it as an open subset of $\mathbb R^3$. Let $E$ be the set of ellipses in ...
- 8,166
12 votes
0 answers
444 views
Is every $\mathbb{Q}$-metrizable space also ultrametrizable?
Suppose that $(X,d)$ is a metric space such that $d$ has range contained in the rationals $\mathbb{Q}$. Question: is it true that $X$ is ultrametrizable? Additional info (edited 30/12/2024): This ...
- 936
1 vote
0 answers
111 views
How would the shape of a circle be different if we change it this way?
What if we define a circle not as the set of points with the distance $R$ from the center, but rather as a set of points which have at least one geodesic path to the center of the length $R$? Will the ...
- 10.4k
-7 votes
1 answer
262 views
Can a space with the following properties exist? Any examples?
Can a space with the following properties exist? Any examples? The number of geodesic paths between any two points is infinite but countable. The infimum of the geodesic path lengths between any two ...
- 10.4k
4 votes
0 answers
66 views
Given a metric space $X$, is there a natural way to view the quasi-isometry group $QI(X)$ as a topological group?
Given a metric space $(X,d)$, we define $QI(X)$ as the set of quasi-isometries $f : X \to X$, modulo the equivalence relation $$ f \sim g \ \ \ \ \text{ if and only if } \ \ \ \sup_{x \in X} \ d(f(x)...
- 970
14 votes
1 answer
790 views
Is there an 'unnatural' topological construction of an algebraically closed field of positive characteristic?
It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly ...
- 19.2k
3 votes
1 answer
205 views
For $\mathbb R^n \times Q \cong \mathbb R^m \times Q $ must $n = m$? ($Q$ is the Hilbert cube)
There are several theorems describing the topology on hyperspaces of convex subsets of $\mathbb R^n$ under the Hausdorff metric. For example Antonyan and Jonard-Pérez prove the space of compact convex ...
- 2,085
0 votes
1 answer
100 views
Metric for measuring linearity of finite set of points in $R^2$
Suppose one has $n > 2$ points in $R^2$, and one wants to measure "how linear" they are. I want a metric such that (a) if all the points are in fact on the same line, the metric gives 1, (...
0 votes
1 answer
147 views
Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
- 79
1 vote
1 answer
160 views
$d(x,y) = \min\{|x_1−y_1|+|x_2−y_2|, 1−|x_1−y_1|+|x_2−(1−y_2)|\}$ defines a metric on $[0,1)\times[0,1]$? [closed]
For $x,y \in [0,1)\times[0,1]$, let $d(x,y)$ be the minimum of $|x_1−y_1|+|x_2−y_2|$ and $1−|x_1−y_1|+|x_2−(1−y_2)|$. Prove or disprove that $d$ is a metric. I was unable to find a counterexample to ...
- 149
0 votes
1 answer
153 views
Embeddings of pseudo metric spaces into seminormed Spaces
There is a theorem stating that every metric space embeds isometrically into $\ell _{\infty}$. My question: is there a generalized result for pseudo metric spaces embedding isometrically into semi-...
- 127
4 votes
1 answer
151 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 ...
- 63.5k
3 votes
0 answers
120 views
Loop space, parametrization equivalence and the issue of giving a topology
This question has been motivated by p.165 of this book. As in the cited link above, we consider the following space of paraemtrized piecewise $C^1$ loops \begin{equation} X:= \Bigl\{ x : [0,1] \to \...
- 3,745
1 vote
0 answers
74 views
Obtaining the geodesic extension property by embedding in a larger space
Suppose $(X,d)$ is a Hadamard space. By considering basic examples like a compact interval in $\mathbb{R}$ or a closed unit ball in Hilbert space, $X$ need not have the geodesic extension property (...
- 163
2 votes
1 answer
216 views
Is completion of measures equivalent to completion of sigma algebras as metric spaces with respect to measures?
An alternative way to get the Lebesgue $\sigma $-algebra $\mathcal{L} $ from the Borel algebra $B$ is to set $E\sim J$ iff $d(E,J):=\lambda(E\mathbin\Delta J)=0$ for $E,J\in B$. Then the completion of ...
- 127
2 votes
1 answer
84 views
Is any submetrizable linear topology linearly submetrizable?
Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$. Is ...
- 5,643
2 votes
1 answer
184 views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$: $$ d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|. $$ Let $\rho$ be the Levy-Prokhorov metric on the ...
- 309