If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?

I know (and am able to prove via Stone-Čech compactification) that the following is correct: Theorem: A metric space is compact if and only if its space of bounded, continuous, …

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 …

Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all proba …

Suppose I have a set with two metrics, which induce distinct topologies, (so neither is contained in the other). There should exist a sequence which converges in both topologies, b …

It is easy to see that the disjoint union $\bigsqcup_i X_i$ of a collection of metric spaces is metrizable, simply by rescaling or chopping off the individual metrics to have diame …

given a metric $d$ an aggregate function $f$ some sets (or multisets or random variables) $X$,$Y$ what do we call: [1] $d(f(X),f(Y)) \leq f( [d(X_0,Y_0) ... d(X_n,Y_n)] )\ \foral …

(I am cross-posting this from math.SE as it seems to be slightly over the top for that site.) I saw in the class the theorem: Suppose $X$ is a separable metric space, and $Y$ is …

In the answers to my previous question, I learned that there are different concepts of distance, that is of distance-like functions with the usual metric being only the most popula …

Has there been study of a generalization of measure spaces along the following or similar lines ? Given a measure space $(X, \Sigma, \mu)$, define for $U\in\Sigma$ a $\mu$-ball o …

I originally posted this question on math.stackexchange (http://math.stackexchange.com/questions/83182/modulus-of-continuity-take-2), but it's been a few days and I haven't receive …

Let $(X,\mu,\mathcal{F})$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras …

As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically, A complete connected …

A metric cone $C$ is a nonempty metric space (whose distance is denoted $d$) together with a map $\cdot\colon \mathbf{R}\times C \mapsto C$ satisfying these axioms: $a\cdot(b\cdo …

Lipschitz maps are defined over metric space as maps $f:(X,d_X) \to (Y,d_Y)$ such that $$ d\left( f(x),f(x^\prime) \right)_Y \le k d(x,x^\prime)_X \ \forall x,x^\prime \in X, $$ wh …