All Questions
Tagged with mg.metric-geometry gn.general-topology
168 questions
1
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59
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Topology on topological spaces
The Gromov-Hausdorff metric makes the set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is ...
- 531
5
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3
answers
286
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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
2
votes
1
answer
95
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Specific distance between sets of points
Let us have closed curve without self-intersections,
initial point $O$ and curve parameter $t$, $0 \leq t \leq t_{\max}$ so $t(O) = 0 = t_{\max}$.
There are two sets of points on the curve, which are ...
- 512
1
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1
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84
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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 $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
9
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2
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471
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Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
- 99
5
votes
1
answer
311
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Quotients in categories of metric spaces
There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
- 16.4k
2
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1
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300
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If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
- 1,149
8
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2
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489
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Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 7,149
0
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1
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115
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Generalized Triangle Inequality for Snowflakes
Let $p>0$ and consider a metric space $(X,d)$. I have recently come across a problem where the space $(X,d^q)$ provides is natural; where $q>1$. However, the triangle inquality break (i.e. it ...
- 169
7
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2
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646
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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
4
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0
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114
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Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
2
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0
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159
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Are there hereditarily square-boxed plane continua?
A plane continuum is a bounded, closed and connected subset of the plane.
A bounding box $B$ for a plane continuum $C$ is
a rectangle $B=[a,b]\times[c,d]$ (including sides and interior)
such that $C$ ...
- 1,375
6
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1
answer
500
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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
5
votes
1
answer
167
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What structure is preserved by pseudo-homeomorphisms of pseudo-Euclidean spaces?
Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb ...
- 41.8k
10
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1
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460
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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
4
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0
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182
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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
7
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1
answer
331
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A metric characterization of Hilbert spaces
In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
- 41.8k
47
votes
3
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3k
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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
3
votes
2
answers
235
views
Uniformly continuous homotopy equivalence
Suppose $M$ and $N$ are complete metric spaces and $f, g: M \to N$ are uniformly continuous maps between them with common modulus of continuity $m$. Further suppose $f$ and $g$ are homotopy equivalent....
- 2,221
11
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1
answer
493
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A topological tree is weakly contractible
Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
- 609
6
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1
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284
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Extending a partially defined metric on a metrizable space
Let $X$ be a metrizable topological space, $A\subseteq X\times X$ a nonempty closed subset which is reflexive, symmetric and transitive, $d:A\to \mathbb{R}_+$ a continuous function that satisfies the ...
- 278
3
votes
1
answer
171
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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
5
votes
0
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171
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Length metrics on covering spaces
This is a question (Exercise 3.30(2)) in the book `Metric spaces of non-positive curvature' written by Bridson and Haefliger.
In the book, there is the following proposition (Proposition 3.28)
Let $p:\...
- 439
7
votes
2
answers
529
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What is the name for a point that is periodic to within $\varepsilon$?
Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...
- 27.7k
6
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0
answers
189
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What is a non-smooth connection?
Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\...
- 63.9k
0
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1
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513
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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 ...
1
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1
answer
124
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A neighborhood $Y$ of a set $X$ such that the line segment connecting any point in $Y$ and its projection to $X$ is contained in $Y$
A direct line from a point $p$ to a set $X$ is a line segment with one endpoint at $p$ and one endpoint in $X$, which is as short as any other line segment from $p$ to $X$. Given a closed set $X$ and ...
- 637
2
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1
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94
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Open covering with bounded diameters [closed]
Here is an interesting puzzle I came across.
I have no idea which tools could be applied to solve it, so the tags may be misleading.
For any $A \subseteq \mathbb{R^n}$ , its diameter is defined by
$$\...
- 29
2
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1
answer
313
views
(Homotopy) colimit and manifold
Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by $D$, over the CW ...
- 37
7
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0
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225
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A weak analogue of smooth manifolds (reformulated)
It is widely known that $C^1$ manifolds are topological spaces locally homeomorphic to Euclidean spaces and possessing $C^1$ chart-converters. They have a tangent space at every point, regarding as ...
- 1,543
3
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0
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64
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Algebraic characterisation of the end space of a proper geodesic space in terms of non-continuous functions
$\DeclareMathOperator\Bf{B_\mathrm{f}}\DeclareMathOperator\Bc{B_\mathrm{c}}\DeclareMathOperator\Cf{C_\mathrm{f}}\DeclareMathOperator\Cd{C_\mathrm{d}}\DeclareMathOperator\Cc{C_\mathrm{c}}$Based on a ...
7
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0
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493
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A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
- 99
3
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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
4
votes
1
answer
110
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Separation of convexity on uniquely geodesic space
A metric $d: X \times X \to [0,\infty)$
is said to be intrinsic provided that the distance between any two points is the infimum of the lengths of
paths joining the points. A space is an inner metric ...
- 2,083
1
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0
answers
70
views
Distance to set defined as subzero level set of a continuous function
I am searching for strategies on how to prove/disprove that scalar functions "capture" the distance to the subzero level set of the same function. (Or what topics to study to become better ...
25
votes
6
answers
2k
views
Are there infinitely many "generalized triangle vertices"?
Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
- 21.2k
0
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0
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177
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On connectedness of the complement
In the application of Runge type theorems on the approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has ...
- 411
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2
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1k
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Can $[0,1]^4$ be partitioned into copies of $(0,1)^3$?
Is there a partition of $[0,1]^4$ such that every member of the partition is homeomorphic to $(0,1)^3$?
More generally, I would like to know for which values of $m$ and $n$ there is a partition of $[0,...
- 18.5k
13
votes
0
answers
818
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Covering number estimates for Hölder balls
Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
- 5,405
7
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1
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242
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Extending continuous injective curves both continuously and injectively
Let $X$ be a topological space.
Let $\gamma:[a,b]\to X$ be continuous and injective.
$\gamma$ is said to be "openly extendable" if there is $[a,b]\subset (a',b')$ and a continuous and ...
- 773
1
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0
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70
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Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
4
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1
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522
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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
3
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0
answers
99
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Condition for: A simple quotient metric induced by surjective map + equivalence relation
Let $X$ be a metric space and let $f:X\rightarrow Z$ be a surjective map onto some set $Z$. Define the pseudo-metric $d_f$ on $Z$ by:
$$
d_f(z_1,z_2)\triangleq \inf_{\underset{f(x_i)=z_i}{x_i\in X}}
\...
- 93
7
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1
answer
195
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Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces?
Is there a countable metric space $U$ such that any countable metric space is bi-Lipschitz equivalent to a subset of $U$? How about $c_{00}(\mathbb{Q})$ where $\mathbb{Q}$ is the rational numbers? ...
- 73
11
votes
2
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305
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Connecting a compact subset by a simple curve
Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative).
Q: Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^...
- 60.5k
5
votes
1
answer
206
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If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a subspace of $H$ close to $F$?
Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ ...
- 5,529
3
votes
0
answers
111
views
Can a path-connected domain be completely surrounded by 4 translates?
Question: Does there exist a compact path-connected set $A\subseteq\mathbb C$ such that:
$A\cap(A+1)=A\cap(A+i)=\emptyset$,
$A\cap(A+1+i)\neq\emptyset$, and
$A\cap(A+1-i)\neq\emptyset$?
Remarks: If ...
- 721
14
votes
1
answer
237
views
Must a path of compact sets in $X$ descend to a path in $X$?
(I am most interested in the case $X=\mathbb R^2$, but of course one could ask the same question for manifolds, or metric spaces in general.)
Let $\text{Com}(\mathbb R^2)$ denote the space of nonempty ...
- 721
7
votes
1
answer
371
views
The space of skew-symmetric orthogonal matrices
Let $M_n \subseteq SO(2n)$ be the set of real $2n \times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x \in \...
- 12.4k
5
votes
0
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144
views
Do products of distance functions separate points?
Let $(X,d)$ be a metric space without isolated points and of diameter $1$. Let $Y=\{y_m\}_{m=1}^{\infty}$ be a dense subset of $X$.
Define $g_0\equiv 1$, and for $m>0$ let $g_m=d(\cdot,y_1)\dotsm d(...
- 5,529