# 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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### When is the internal covering number of a metric space monotonic?

Given a radius $r > 0$, the internal covering number of a subset $T$ of a metric space $(X, d)$ is denoted $N_r(T)$ and is defined to be the smallest number of balls of radius $r$ (under $d$) with ...
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### Homeomorphism between $L^p$-spaces on metric spaces and $L^p$-spaces on Euclidean space

Setup: Fix $p \in [1,\infty)$. Let $(X,d_X,x_0)$ and $(Y,d_Y,y_0)$ be complete pointed metric spaces and $\mu$ be Borel. Let $E^n,E^D$ be Euclidean spaces of respetive dimensions $n$ and $D$ and ...
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### When are Carnot groups negatively curved and homeomorphic to Euclidean space

When are Carnot groups complete and negatively curved (in the sense of $CAT(\kappa)$ spaces)?
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### Proving that family of sets has non-empty intersection

Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already: $S$ is set of measurable ...
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### Closure Wasserstein for pointmasses

Suppose that $(X,d)$ is a metric space which is (not necessarily) complete and let $W_1(X)$ denote the Wasserstein metric space on $(X,d)$. Let $\{\delta_x\}_{x \in K}$ is a collection of degenerate ...
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### Analogy between metric space completion and algebraic closure

I've noticed some similarities between the story of completing a metric space and taking algebraic closure of a field. My question is whether these two stories can be generalized. Metric space Fix a ...
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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 $G(X,d)$ ...
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(...