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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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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, \...
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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 ...
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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 ...
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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 ...
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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" ...
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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 \...
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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: $$...
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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 ...
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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 ...
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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 ...
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$\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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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}_{\...
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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 $T_{f(p)}Y$ ...
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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 ...
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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 “...
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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 ...
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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 ...
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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\...
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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}$? ...
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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\}$....
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
James E Hanson's user avatar
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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 ...
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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, (...
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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 $G$...
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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 ...
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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-...
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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 ...
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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 \...
Isaac's user avatar
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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 (...
E G's user avatar
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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 ...
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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 ...
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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 ...
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