# Tagged Questions

**3**

votes

**1**answer

95 views

### Lipschitz function with somewhere dense image

Let $Q=[-1,1]^2$ denote the unit square and let $f:Q\to Q$ be a Lipschitz function such that for any ball $B(a,r)\subset Q$ with radius $r$, the width of the image $f(B(a,r))$ is at least $cr$ for ...

**9**

votes

**1**answer

196 views

### Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?

Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...

**9**

votes

**0**answers

157 views

### Multiplicity of ball covering

Background. My questions are motivated by the following:
A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete
and Computational Geometry, 8 (1992) 109-130) considered the ...

**2**

votes

**0**answers

141 views

### Comparing two metrics on the space of infinite sequences and relating open and closed sets

Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it
$$
d(u,v) := \frac{1}{2^r} \mbox{ with } ...

**4**

votes

**1**answer

200 views

### Hausdorff metric on C[0,1]

Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in ...

**5**

votes

**1**answer

658 views

### A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite ...

**4**

votes

**1**answer

194 views

### n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...

**4**

votes

**3**answers

522 views

### Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...

**6**

votes

**2**answers

469 views

### Homotopy problem for infinite dimensional topological space II

This post here is a specification of this post.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying :
$X_{n}$ have topological dimension $n$.
$X_{n+1}$ is ...

**7**

votes

**1**answer

552 views

### What is the shape of mathematical universe?

Shape? At the usual mathematical literature when we can discuss about the shape of a "space" that we have a kind of "topography" on it. For example a topology, metric, geometry, etc.
Note that for ...

**6**

votes

**1**answer

228 views

### Does a metric refine the weak-* topology on a dual space?

Let $X$ be a topological affine space over $\mathbb C$, with no additional assumptions. Let $X^*$ denote its dual space of continuous affine functionals $X \to \mathbb C$, equipped with the weak-$*$ ...

**7**

votes

**1**answer

275 views

### Constructing Metrics for specific Topological Spaces, and Refinements of the Cantor-Space in particular

I have a Problem in general, given some some Topological Space $(X, \tau)$ from which I know it is metrisable, how can I find a metric (that is at best in some sence constructive and easy, at the very ...

**4**

votes

**2**answers

304 views

### Is there a contractible bounded homogeneous space?

Let us phrase the question in the title in more detail: I wonder if there exists a metric space $X$ which has at least two points, has finite diameter (in the sense that there is an upper bound for ...

**0**

votes

**1**answer

175 views

### The pth power of a distance function is twice continuously differentiable, for $p>2$?

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...

**6**

votes

**1**answer

746 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

203 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 ...

**8**

votes

**1**answer

364 views

### Uniform Embedding into Euclidean Space

Given a locally compact, separable, metric space $X$.
When does $X$ uniformly embed into some Euclidean space?
This means, when does there exist some integer $n$ and a closed subset ...

**1**

vote

**3**answers

368 views

### Metric properties for $d:X\times X\times…X\rightarrow\mathbb R$ [closed]

Let us define $d:X^n\rightarrow\mathbb R$. How can we define metric properties such as symmetry, triangle inequality equivalent property etc for such a function?

**36**

votes

**7**answers

3k views

### Is there an algebraic approach to metric spaces?

It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there ...

**3**

votes

**1**answer

298 views

### Least cardinality of a set of points in the plane

What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...

**2**

votes

**1**answer

277 views

### showing uniformly continuous

Let $(X,d)$ be a metric space and $(a_n)$ be a sequence of distinct points in $ X$ such that each $a_n$ is a limit point of $X$. If $U_n$ 's are mutually disjoint open neighbourhoods of $a_n$ in $X$. ...

**6**

votes

**6**answers

709 views

### Bijective function on a dense set

Suppose X is a complete metric space, and $f:X↦X$ a continuous surjective function. Let D be a dense set. Suppose $f:D↦D$ is injective and $f^{-1}(D)=D$.
Is $f$ injective ?
Is there a family of ...

**0**

votes

**0**answers

175 views

### Decompose a set into sets of Hausdorff-dimension n-1

Assume we can decompose a set $A$ in $\mathbb{R^n}$ of Hausdorff-dimension n into sets $(A_t)$ $t\in [0,1]$ of Hausdorff-dimension n-1 whose n-1-dimensional volume is known (for example is zero).
...

**4**

votes

**1**answer

259 views

### Is there a name for the class of metric spaces such that the closure of the open ball of radius $r$ around each point $x$ is the set of elements $y$ such that $d(x,y)\leq r$ ?

Let $(X,d)$ be a metric space, let $B(x,r)$ be the open ball of radius $r$ about $x$ and $N(x,r)$ be the set of elements $y\in X$ such that $d(x,y)\leq r$. It is well-known that it is not always true ...

**1**

vote

**2**answers

479 views

### Injective Function on a Dense Set

This is a topological question that came up tangentially to some material I was working on. Suppose $X$ and $Y$ are complete metric spaces and $D$ is a dense subset of $X$. Let $f:D\mapsto Y$ be a ...

**3**

votes

**2**answers

437 views

### Shortest “painting” of the sphere

Let $S$ be the sphere in $\mathbb{R}^3$ and $C:[0,1]\to S$ a continuously differentiable curve on $S$. Let $T:[0,1]\to\mathbb{R}^3$ denote the tangent vector of $C$. Let $P(t)$ be the plane containing ...

**2**

votes

**1**answer

706 views

### A density condition for metric spaces

I have encountered the following property. Can anybody tell me if it already exists in literature and/or is equivalent/similar to other well-known properties?
Property: $(X,d)$ metric space. For ...

**1**

vote

**1**answer

604 views

### Lebesgue covering dimension

Roughly from wikipedia: The covering dimension of a topological space $X$ is defined to be the minimum value of $n$ such that every finite open cover of $X$ has a finite open refinement in which no ...

**13**

votes

**2**answers

942 views

### Example of a compact homogeneous metric space which is not a manifold

A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd ...

**13**

votes

**4**answers

1k views

### Decidability of tiling R^2

Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane?
I know the ...

**16**

votes

**3**answers

3k views

### Nonseparable example in dimension theory?

Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to ...

**1**

vote

**1**answer

558 views

### What are the topological properties of the metric space retained (inherited) for its completion

Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained ...

**11**

votes

**3**answers

670 views

### Can Lipschitz maps increase the Lebesgue dimension ?

Given a map $f:X\rightarrow Y$ of compact metric spaces, such that there is a $C\in \mathbb{R}$ with $d(f(x),f(x'))\le C\cdot d(x,x')$.
Does this already imply, that the Lebesgue dimension of $f(X)$ ...

**5**

votes

**1**answer

454 views

### When completion of locally compact length space is locally compact?

As far as I know the answer to the question:
"Is it true that a completion of a locally compact length space is locally compact?" - Negative.
Does anybody know some metric and/or topological ...

**10**

votes

**2**answers

637 views

### Are there space filling curves for the Hilbert cube ?

There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...

**1**

vote

**1**answer

412 views

### Сomplete homogeneous space which is not locally compact

It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?

**3**

votes

**1**answer

541 views

### Do continuous maps give continuity in the 'topology' of Hausdorff distance?

I was reading this question:
limiting behaviour of converging loops on a torus
And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their ...

**5**

votes

**2**answers

513 views

### Quantitative questions about the size of a finite epsilon net

Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of ...

**12**

votes

**4**answers

1k views

### What is the “right” universal property of the completion of a metric space?

I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes:
Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...

**-6**

votes

**4**answers

562 views

### What is the max number of points in R^3, interconnected by generic curves?

The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of ...