41
votes
Accepted
What happens if you strip everything but the “between” relation in metric spaces
There is a wide body of work on this in connection with the classic De Bruijn–Erdős theorem.
De Bruijn–Erdős Theorem. Every set of $n$ points in the
plane (not all lying on the same line) ...
- 32.1k
37
votes
Is there a "universal" connected compact metric space?
There is no such continuum. See
Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205.
Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called ...
- 7,193
25
votes
Accepted
A funny metric over $\mathbb{N}$
This metric is mentioned in the Encyclopedia of distances (Chapter 10.3), written by Michel Marie Deza and Elena Deza. Here is the relevant paragraph:$\newcommand{\lcm}{\operatorname{lcm}}$
Let $\...
- 8,401
23
votes
When does a metric space have "infinite metric dimension"? (Definition of metric dimension)
Well, if $M$ has a metric basis $\{b_1, \ldots, b_{n+1}\}$ then the map $$x \mapsto (d(x,b_1), \ldots, d(x,b_{n+1}))$$ is a continuous injection from $M$ into $\mathbb{R}^{n+1}$. So, for example, no ...
- 42.8k
18
votes
What happens if you strip everything but the “between” relation in metric spaces
I'm reminded of W.A. Coppel's book which looks at these kinds of structures from a slightly different vantage point, namely closure systems. I can't actually find the book right now, but here's a ...
- 3,793
18
votes
Accepted
BCT equivalent to DC
You can find it, amongst other places in my write up:
Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice.
If you need a source to cite, my money is on ...
- 39.7k
17
votes
What is the structure preserved by strong equivalence of metrics?
Metrics are strongly equivalent if the identity mapping $Id:(X,d_1)\to (X,d_2)$ is bi-Lipschitz. They preserve the class of Lipschitz mappings.
Roughly speaking classical topology deals with notions ...
- 28k
17
votes
Accepted
Is the topology generated by this weaker notion of a metric necessarily metrisable?
For a loose metric $d$ as above, we can consider the function
$$d_1(x,y):=\sup\{|d(x,z)-d(y,z)|;z\in X\}.$$
It is easy to verify that $d_1$ is a metric, and $d(x,y)\leq d_1(x,y)\leq\rho(d(x,y))$ for ...
- 10.6k
16
votes
Accepted
Partition of unity without AC
The proofs rely, in the background, on Urysohn's Lemma, which follows from the Principle of Dependent Choices but is not provable without some Choice. It is false in the ordered Mostowski model, see
...
- 11.4k
15
votes
Accepted
Does a compact contractible metric space have a point that is fixed by all isometries?
There are finite groups that act smoothly on a disk without a global fixed point. You can arrange the metric to be isometric, e.g. via the Mostow-Palais embedding theorem, which equivariantly and ...
- 29.1k
14
votes
Accepted
On the Large Cardinal Strength of Normal Moore Space Conjecture
If $\text{NMSC}$ is consistent, then so is $\text{NMSC}+\text{"there are no strongly inaccessible cardinals"}$.
This is because if $V \models \text{NMSC}$, then $V_\kappa \models \text{NMSC}$ for ...
- 18.5k
14
votes
Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
Let $X$ be the unit circle and let $Y$ be the Koch snowflake, both with euclidean metric inherited from $\mathbb{R}^2$. There is a continuous homeomorphism from $X$ onto $Y$, but there is no ...
- 42.8k
13
votes
Euclidean tangent cone implies Riemannian manifold
To provide some context the subsets of a Euclidean space that can be approximated by affine planes on every scale are known as Reifenberg-flat sets after E. R. Reifenberg who proved in the 1960s that ...
- 29.1k
12
votes
Accepted
Totally bounded spaces and axiom of choice
The issue here is that a metric space might not have non-trivial (read: not eventually constant) Cauchy sequences. For example, if the underlying space is a Dedekind finite set.
Indeed it is ...
- 39.7k
12
votes
Accepted
A reinterpretation of the $abc$ - conjecture in terms of metric spaces?
$d_2$ is indeed a metric. Abbreviating $\gcd(m,n)$ to $(m,n)$, we need to show that
\begin{align*}
1-\frac{2(a,c)}{a+c} &\le 1-\frac{2(a,b)}{a+b} + 1-\frac{2(b,c)}{b+c}
\end{align*}
or ...
- 12.8k
12
votes
BCT equivalent to DC
Wikipedia article on Baire category theorem and several other sources mention this paper: Blair, Charles E. The Baire category theorem implies the principle of dependent choices. Bull. Acad. Polon. ...
- 4,713
12
votes
Accepted
There exists differentiable curves arbitrarily close to the continuous ones
It turns out that something much more general is true and can be found in the literature.
Theorem [Thm 3.3, Hirsch, Differential Topology] Let $M$ and $N$ be $C^s$-manifolds (with boundary), $1\le s\...
- 2,806
11
votes
When does a metric space have "infinite metric dimension"? (Definition of metric dimension)
Having infinite metric dimension is not bi-Lipschitz-invariant.
On the real line, consider the two metrics
$$
d(x,y)=\min(|x-y|,1)\quad\text{and}\\
\delta(x,y)=\arctan(|x-y|).
$$
The two metrics both ...
- 8,086
11
votes
Accepted
Spreading $n$ points in $\{0,1\}^n$ as far as possible
If I understood the question correctly, what you're asking is related to the maximum distance of binary codes with large minimum distance $d\geq s$.
In coding theory, $A_q(n,d)$ is defined as the ...
- 10.4k
11
votes
Accepted
Examples of metric spaces with measurable midpoints
We will use the Kuratowski–Ryll-Nardzweski selection theorem:
Let $(\Omega, \mathscr{F})$ be a measurable space. Let $E$ be a Polish space. Let $\Gamma$ be a set-valued function from $\Omega$ to $E$;...
- 41.1k
11
votes
A generalization of metric spaces
My PhD thesis was on this topic, focusing mostly on a combinatorial approach (rather than topological). So what I write below is directed toward Question 1.
I call a structure $\mathcal{R}=(R,+,\leq,0)...
- 3,274
11
votes
Smooth Urysohn's lemma on Fréchet spaces
A bump function on a Banach space $X$ is a non-zero function with bounded support. Existence of a bump function of a given smooth regularity has, of course, strong immediate consequences --by ...
- 60.5k
11
votes
Accepted
Is it possible to prove that any two points of a convex complete metric space are connected by some metric segment without the axiom of choice?
If dependent choice fails then there is a convex complete metric space metric space with two points not connected by a segment. I'm not yet sure about the converse. It seems clear that $\mathsf{DC}_{\...
- 12.4k
10
votes
How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?
Hu Xiyu. Even though yours is a question from four years ago, I want to bring to your attention my recent paper “Gromov-Hausdorff distance between spheres”( https://arxiv.org/abs/2105.00611 ) ...
- 469
10
votes
Accepted
A generalization of metric spaces
It follows from your assumptions that for $a<b\in L$ there is a unique $c$ such that $a+c=b$ and that $L$ is a cancellative monoid: $a+c=b+c$ implies $a=b$. Also addition preserves the order. A ...
- 1,089
10
votes
In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?
It seems to me that you can show that no infinite-dimensional separable Banach space $X$ is P-complete as follows. Pick any bounded separated sequence $\{x_n\}_{n=1}^\infty$ in $X$ and pick a dense ...
- 4,878
10
votes
Accepted
In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?
That every Banach space is contained in a $P$-complete Banach space follows immediately from the following
Theorem.
Let $X$ be a Banach space. Then there exists a Banach space $Y$ containing $X$ in ...
- 31.5k
10
votes
Accepted
Relationship between doubling constant of a metric space and of a metric measure space
Apart from the obvious counterexample of the measure being $0$, if $(X,d,m)$ is doubling in the sense of metric measure spaces it will be doubling in the sense of metric spaces.
Consider a ball $B(x,r)...
- 10.6k
Only top scored, non community-wiki answers of a minimum length are eligible