All Questions
23 questions
47
votes
3
answers
3k
views
A metric characterization of the real line
Is the following metric characterization of the real line true (and known)?
A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real ...
- 41.8k
18
votes
2
answers
2k
views
Which platonic solids can form a topological torus?
8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...
- 453
12
votes
1
answer
559
views
Square lying on moving chord of a simple closed curve
Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate a point $c_1$ on the left and $c_2$ on the right side to the chord ab, such that $ac_1bc_2$ ...
- 415
10
votes
1
answer
460
views
An incomplete characterisation of the Euclidean line?
We say that a metric space $(X, d)$ is a Banakh space if for every $\rho \in \mathbb{R}_{> 0}$ and every $x \in X$, there are
$a,b \in X$ such that $\{y \in X \, \vert \, d(x, y) = \rho\} = \{a, b\}...
- 7,893
10
votes
1
answer
346
views
A forked plane continuum
I came up with this question while trying to solve the following MO one:
Does every connected set that is not a line segment cross some dyadic square?
Suppose $C$ is a plane continuum (i.e. a ...
- 1,375
10
votes
1
answer
1k
views
Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane
Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$).
Let $x \in \Sigma$, and suppose you have the following: for every $r<1$,
the open ...
- 353
9
votes
1
answer
726
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 $Y\subset\...
- 3,497
7
votes
2
answers
646
views
A generic metric on $X\cup\mathbb Z$
$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$.
Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that
$d(x,y)=d_X(x,...
- 41.8k
6
votes
1
answer
500
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 41.8k
6
votes
1
answer
450
views
Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...
- 41.8k
5
votes
3
answers
286
views
On a metrized $n$-dimensional manifold $X$, does every $x \in X$ have a small ball $B_\delta(x)$ that is homeomorphic to $\mathbb R^n$?
Suppose that $X$ is an $n$-dimensional topological manifold that is also metrizable, and hence equipped with some metric that induces the topology.
For every point $x \in X$, let $B_\delta(x)$ be the ...
- 5,327
5
votes
1
answer
1k
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 ...
- 3,230
5
votes
1
answer
452
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 ...
- 98
4
votes
1
answer
522
views
Can every manifold be represented as a quotient
My question is "inspired" by the uniformization theorem for Riemmannian surfaces and this post.
Suppose that $X$ is connected (finite-dimensional) topological manifold without boundary. ...
- 5,405
4
votes
0
answers
182
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 41.8k
3
votes
1
answer
171
views
Spaces satisfying a strong Cartan-Hadamard theorem
Let $(X,d)$ be a connected geodesic metric space. When does there there exists a covering map $\pi:H\rightarrow X$ which is a local-isometry where $H$ is either a Hilbert space or a Euclidean space?
...
- 364
3
votes
0
answers
177
views
When do Polish spaces admit complete metric making them $\mathrm{CAT}(\kappa)$?
Question
$\DeclareMathOperator\CAT{CAT}$Let $X$ be a Polish space. When are there known conditions under which $X$'s topology can be metrized by a metric $d$ such that $(X,d)$ is a:
$\CAT(\kappa)$ ...
- 195
3
votes
0
answers
118
views
Weak contractibility of some infinite dimensional metric spaces
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 type$^4$, ...
2
votes
1
answer
226
views
Moving chord on the simple closed curve
Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate point $c$ on the left (or right) side to chord $ab$ such that $\angle acb = 90^{\circ}, ac=...
- 415
2
votes
0
answers
77
views
Dense embeddings into Euclidean space
The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
- 5,405
1
vote
1
answer
84
views
Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
0
votes
1
answer
513
views
Distance between two points using triangulation
Suppose we have two points $p_1$ and $p_2$ in a metric space with unknown dimensionality, with no way to directly compute the distance between them, e.g. no coordinates.
Say we can randomly sample a ...
-4
votes
4
answers
677
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 it....
- 555