The metric-spaces tag has no usage guidance.

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### The Schwartz space is not normable

The Schwartz space of rapidly decreasing function (as well as their derivatives) on $\mathbb R^n$ is a Fréchet space, whose (metric complete) topology is given by the usual countable family of ...

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109 views

### Minkowski spacetime in Newman Penrose formalism

I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere:
I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's ...

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286 views

### Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...

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146 views

### How to change the given metric if we want to add few extra isometries?

I have a Hilbert Space $X$ and a group $G$, which consists of bounded linear self-bijections of $X$ (if it helps, this group has a locally compact, but not compact topology). Is there a canonical way ...

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127 views

### Is there an $\infty$ version of the Wasserstein distance between two distributions?

If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the $p$-th Wasserstein distance between the two of them is defined as $$W_p(\mu,\nu) = ...

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225 views

### Spaces that can't be embedded in the plane

If $X$ and $Y$ are topological spaces, let us write $X \preceq Y$ whenever $X$ embeds in $Y$.
Earlier today, I asked the question:
Is this a well-quasi-order on the completely metrizable spaces?
...

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**1**answer

90 views

### Conditions for a set being closed under taking complement of a ball twice

Given a subset $S$ of a finite metric space $F$ with a distance function $d(,)$ and a number $\delta > 0$ let $N_\delta(S) = \{x \in F| d(x,S)\ge \delta\}$. Is there a characterization of ...

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94 views

### Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies :
$d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) ...

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74 views

### Equicontinuity of $\{f_{2n}\circ f_{2n-1}\}$

Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by ...

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346 views

### Terminology for metrics?

For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for ...

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162 views

### A generalization of SOCA

Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...

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807 views

### Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the ...

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464 views

### Quotient of metric spaces

Let $(X,d)$ be a compact metric space and $\sim$ an equivalence relation on $X$ such that the quotient space $X/\sim$ is Hausdorff. It is well known that in this case the quotient is metrizable. My ...

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265 views

### Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...

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78 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 ...

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110 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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38 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 ...

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328 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\ ...

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234 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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263 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) ...

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331 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 ...

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578 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 $y$?

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118 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 ...

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624 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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68 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 ...

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217 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 ...

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

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### 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 ...

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63 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 ...

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### 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: ...

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533 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] ...

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554 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 ...

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259 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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118 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 $B(x_2,r_2)\subset B(x_1,r_1)$?

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462 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 ...

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165 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)). ...

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74 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 ...

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193 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 ...

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241 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 ...

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203 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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### 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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### 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:
...

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### 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 ...

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371 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 ...

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243 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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### 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?
...

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### 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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### 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 ...

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### 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 ...

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### 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 ...