The metric-spaces tag has no wiki summary.

**1**

vote

**3**answers

190 views

### What is the most ``diverse'' $k$-subset of $[0, 1]^m$?

Given a non-negative integer $m$, let $\Omega_m$ denote the set of vectors $\omega = (\omega_1, \dots, \omega_m) \in [0, 1]^m$ such that $\sum_i{\omega_i} = 1$.
The set $\Omega_m$ is together with a ...

**1**

vote

**0**answers

78 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 $d: X \times X \to [0,\infty]$ ...

**2**

votes

**2**answers

169 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 ...

**2**

votes

**0**answers

47 views

### A construction with Hyperspace of continums

Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a ...

**1**

vote

**2**answers

351 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: ...

**5**

votes

**1**answer

365 views

### sets without perfect subset in a non-separable completely metrizable space

Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements?
[1] ...

**2**

votes

**2**answers

288 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 ...

**5**

votes

**1**answer

166 views

### Completely Metrizable Space and Baire Theorem

Is well know that completely metrizable spaces are Baire's spaces. Reciprocally, if $X$ is a Baire's metric space, then $X$ is completely metrizable?

**-3**

votes

**3**answers

100 views

### Metric-space with a ball inside a smaller ball [closed]

Could you tell me an example to an $(X,\varrho)$ metric-space with balls $B(x_1,r_1)$ and $B(x_2,r_2)$ where $r_1<r_2$ but also $B(x_2,r_2)\subset B(x_1,r_1)$?

**3**

votes

**1**answer

221 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 ...

**1**

vote

**1**answer

149 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)). ...

**2**

votes

**0**answers

57 views

### Metric space has a basis countably locally finite

it is know that all metric space has a basis countably locally finite and this result is proved by using axiom of choice. Then, the natural question is: is possible to prove this result without using ...

**1**

vote

**2**answers

150 views

### Metrization of spaces of functions

Let $M$ and $N$ be topological spaces. Are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?
For $M$ compact and $N$ a metric ...

**4**

votes

**1**answer

183 views

### Inducing metric spaces

Let $f\colon \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$ be a function. We say that $f$ has the property of inducing metric spaces, whenever for all metric space $(X,d)$, $(X, f \circ d)$ is also a ...

**1**

vote

**1**answer

160 views

### A measure of closeness to a discrete set in a metric space

Consider a metric space $(M,d)$ and consider a collection of points $X_n := \{x_1,\dots,x_n\} \subset M$. Let
$$
N_\epsilon(y;X_n) := | \{ x \in X_n: d(x,y) \le \epsilon \}|
$$
where the RHS is ...

**8**

votes

**0**answers

334 views

### Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...

**14**

votes

**4**answers

680 views

### Metrics for lines in $\mathbb{R}^3$?

I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$.
Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$.
Ideally I would like these properties:
...

**0**

votes

**1**answer

112 views

### Is there any result concerning on the metric dimension of inverse limit?

To be specific, my question is as follows:
Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold
$\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some ...

**4**

votes

**1**answer

349 views

### “monotone” homotopy?

This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and ...

**7**

votes

**1**answer

233 views

### Unbounded metrics on groups

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

**3**

votes

**3**answers

334 views

### Compactness of sigma-algebra for the $L^1$ metrics

Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space?
...

**9**

votes

**1**answer

354 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 ...

**8**

votes

**3**answers

775 views

### Axiom of Choice and continuous functions

Do you know if the following statement is an equivalent form of the axiom of choice or not?
If $X$ is a compact metric space, then every continuous function $f: X \longrightarrow \mathbb{R}$ is ...

**7**

votes

**0**answers

199 views

### Construct a topologically $\infty$-dimensional separable metric space.

But don't assume knowledge of any topological dimension theory. Here is a specific approach (an open problem):
Does there exist a separable metric space $X$ such that the following two conditions ...

**6**

votes

**1**answer

705 views

### reference for “X compact <=> C_b(X) separable” (X metric space)

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, real-valued ...

**6**

votes

**1**answer

197 views

### Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as “natural” / “induced”?

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 probability distributions ...

**5**

votes

**2**answers

362 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 ...

**2**

votes

**2**answers

220 views

### Two metrics and a sequence converging to two points. [closed]

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, but to different ...

**8**

votes

**0**answers

495 views

### In ZF, when is a disjoint union of metrizable spaces metrizable?

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 diameter at most one, and ...

**1**

vote

**0**answers

134 views

### Z-sets in the Hilbert cube

If $(X,d)$ is a metric space, then we say that a closed subset $A$ of $X$ is a z-set if for each number $k\gt 0$ there is a continuous map $f_k$ from $X$ into $X-A$ such that $d(x,f_k(x))\lt k$.
I ...

**4**

votes

**0**answers

392 views

### What relates to measure spaces as topological spaces relate to metric spaces ?

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 of radius $r$ of $U$ ...

**9**

votes

**3**answers

662 views

### What makes a distance?

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 popular and important one.
...

**3**

votes

**1**answer

103 views

### Independence of the axiomatics of metric cones

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:
$a\cdot(b\cdot x) = (ab)\cdot x$ ...

**4**

votes

**1**answer

633 views

### Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?

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 Riemannian manifold ...

**10**

votes

**1**answer

1k views

### Modulus of Continuity

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 received any correct ...

**4**

votes

**2**answers

338 views

### Is the Hausdorff metric on sub-$\sigma$-fields separable?

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) of $\mathcal{F}$ ...

**4**

votes

**1**answer

416 views

### Equivalent metrics on Fréchet spaces and Lipschitz maps

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

**3**

votes

**0**answers

257 views

### For METRIZABLE spaces, do the Banach classes and Baire classes coincide?

In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...

**11**

votes

**1**answer

580 views

### Converse to Banach’s fixed point theorem for ordered fields?

Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := ...

**8**

votes

**2**answers

474 views

### What does the space induced by this unusual metric(?) on R/Z look like?

The motivation for this question comes from music theory. Dmitri
Tymoczko models "good" voice leading as minimizing distance between
pitches in successive chords. While this theory works well for ...

**0**

votes

**1**answer

398 views

### Classes of metric spaces with additional structure [closed]

As is often the case in mathematics there is an option of studying a more general topic but this comes with a price of losing some interesting properties which are only present in the more specialized ...

**10**

votes

**3**answers

852 views

### Universal sets in metric spaces

(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 a polish space ...

**0**

votes

**1**answer

488 views

### How the distance between sets is called?

Hello,
I've recently write down some measure for sets and now I wonder how it is called or where it is described?
The measure itself is the following:
Let $A$ & $B$ -- two sets of values from a ...

**10**

votes

**4**answers

877 views

### When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...

**9**

votes

**3**answers

831 views

### Spaces with a quasi triangle inequality

How do you call a space with a function which is symmetric, non negative, positive definite and which satisfies a quasi-triangle inequality:
$d(x,z) \leq C( d(x,y)+d(y,z) )$
for all $x,y,z$ and some ...

**9**

votes

**1**answer

1k views

### metric on the space of real analytic functions

Hello,
this question may be simple but I couldn't find a reference.
Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C_b^{\omega}(\Omega,F)$ be the vector space of ...

**3**

votes

**1**answer

153 views

### cardinality of local bases in the non-standard reals

Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of ...

**7**

votes

**2**answers

817 views

### End point compactification for metric spaces

Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...

**15**

votes

**1**answer

704 views

### covering a separable metric space by small balls

Let $(X,d)$ be a separable metric space. Can $X$ always be covered by a sequence of balls $B(x_i,r_i) (i=1,2,\dots)$ s.t. radii $r_i$ tend to 0?

**9**

votes

**1**answer

388 views

### Are packing-homogeneous spaces homogeneous?

Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...