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0
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0answers
39 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
2answers
145 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 ...
0
votes
0answers
63 views

Isometry groups acting transitively [migrated]

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
2
votes
0answers
45 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 ...
0
votes
2answers
338 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: ...
4
votes
1answer
310 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
244 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
150 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
3answers
98 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 $B(x_2,r_2)\subset B(x_1,r_1)$?
2
votes
1answer
188 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
1answer
145 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
0answers
56 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
2answers
138 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
175 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
1answer
150 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 ...
9
votes
0answers
298 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
4answers
644 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
109 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
342 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
1answer
232 views

Unbounded metrics on groups

If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?
3
votes
3answers
309 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
1answer
348 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
3answers
757 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 ...
6
votes
0answers
195 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
1answer
612 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
1answer
194 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
340 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
2answers
214 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
0answers
479 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
0answers
132 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
0answers
374 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
3answers
658 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
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
617 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
1answer
981 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
2answers
329 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
1answer
406 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
0answers
254 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
1answer
575 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
2answers
466 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
1answer
395 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
3answers
832 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
1answer
482 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 ...
8
votes
3answers
799 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
1answer
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
1answer
151 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 ...
6
votes
2answers
776 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 ...
14
votes
1answer
695 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
1answer
383 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 ...
8
votes
2answers
250 views

Are all homogeneous metric spaces bihomogeneous?

Let (X,d) be a metric space such that for all points p and q in X, there exists an isometry f such that f(p) = q. Does it follow that for all points p and q in X, there exists an isometry f such that ...