# 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) $d(x,y)+d(y,z) \geq d(x,z)$.

31 questions
Filter by
Sorted by
Tagged with
877 views

### If all balls at $x$ and $y$ are isometric is there an isometry sending $x$ to $y$?

Let $(X,d)$ be a metric space and $x,y \in X$. Assume that for all $r > 0$ the balls $B_r(x)$ and $B_r(y)$ are isometric. Is it true that there exists an isometry of $X$ sending $x$ to $y$?
400 views

420 views

### Probability that a random distance function is metric

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...
2k views

### 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 ...
1k 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 ...
998 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: ...
505 views

### 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 ...
752 views

### 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 ...
311 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 ...
255 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 ...
217 views

### If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post). Basically following some ideas of W. Lawvere (but not his ...
113 views

### First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric

This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions. We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
75 views

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 $$E(G(X,d)) = \big\{\{x,y\}:x\neq y\in X \text{ and } d(... 1answer 184 views ### Open set of geodesics implies the set of starting points is open Let X be a complete and separable metric space, let G(X) \subset C([0,1],X) be the space of continuous curves from [0,1] to X with constant speed, i.e.$$ d(f(t),f(s)) = |t-s| d(f(0), f(1)). $$... 1answer 92 views ### Continuous inclusion of metric spaces of smaller capacity If (X,d_X) is a compact metric space, and (Y,d) is another metric space. Moreover, suppose that the metric capacity of (Y,d) is at-least that of (X,d_X), that is$$ \kappa_X(\epsilon)\leq \...
This is a follow-up to this post. Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by  \kappa_X(\epsilon)\triangleq\sup\left\{ k : \exists x_0,\dots,x_k \...
### Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following: For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set \$\mathcal{F}\...