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62 views

Nearly injective Banach spaces

There was a problem about nearly injective metric spaces posed by Aronszajn and Panitchpakdi which I actually solved in the past but it still remains open (as long as I know) for the Banach spaces--so ...
5
votes
1answer
93 views

Continuity of central point operation

Stanisław Mazur and Stanisław Ulam, in their joint paper, characterized the mid-point $\ \frac{a+b}2\ $ in a Banach space in pure metric terms (without algebra). This allowed them to show that any two ...
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0answers
30 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
4
votes
2answers
263 views

Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ ...
2
votes
2answers
206 views

A metric associated with a continuous surjective map $f:X\to Y$

Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another metric $d_{f}$ on $Y$ With $$ d_{f}(y_{1},y_{2})=Hd(f^{-1}(y_{1}), ...
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0answers
445 views

A metric on $S^{2}$ [closed]

Edit:Can this new version of this question be answered with the method of same comments to the previous version? Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel ...
8
votes
1answer
175 views

Gromov-Hausdorff convergence for non-compact metric spaces

Let $(X_i,p_i)$, $(X,p)$ be pointed connected proper metric spaces (i.e. the closures of balls are compact). Are the following two statements equivalent? $\forall r > 0: \bar{B}_r(p_i) ...
13
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1answer
300 views

If all balls around two points are isometric… — manifold version

This question is a natural follow-up of this other question, asked earlier today by wspin. Let's say that a metric space $(X,d)$ has two poles if: there are two distinct points $x$, $y$ such that ...
18
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2answers
490 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 $y$?
1
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1answer
112 views

Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
18
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2answers
499 views

Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...
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0answers
58 views

Cover a set with balls centered at smooth functions (Ascoli theorem)

Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric. Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the ...
1
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3answers
204 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
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0answers
82 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
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2answers
179 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
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0answers
60 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
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2answers
400 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
1answer
406 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
2answers
399 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
1answer
204 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?
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votes
3answers
109 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
1answer
300 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
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1answer
152 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
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0answers
61 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
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2answers
167 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
1answer
211 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
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1answer
193 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 ...
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0answers
473 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 ...
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4answers
857 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
1answer
114 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
1answer
360 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
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1answer
239 views

Unbounded metrics on groups

If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?
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3answers
370 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? ...
10
votes
1answer
377 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 ...
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3answers
797 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
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0answers
210 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 ...
7
votes
1answer
929 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 ...
8
votes
4answers
375 views

Which metric spaces have this superposition property?

Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$. Say that a metric space $X$ has the superposition property (my terminology) ...
6
votes
1answer
222 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
2answers
430 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 ...
3
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2answers
243 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 ...
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0answers
516 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 ...
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0answers
138 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 ...
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0answers
455 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 $U$ ...
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3answers
667 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
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1answer
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
1answer
662 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
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1answer
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 ...
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2answers
353 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 $\mathcal{F}$ ...
5
votes
1answer
436 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 ...