Intereting Posts

$f(z)=\int_1^\infty e^{-x}x^z\,dx$ is complex analytic
Count of lower and upper primitive roots of prime $p \equiv 3 \bmod 4$
Proof of a theorem saying that we can make a measurable function continuous by altering it by a set of arbitarily small measure
C* algebra inequalities
Matrix generated by prime numbers
Number of ways to separate $n$ points in the plane
How to prove that $(A \lor B) \land (\lnot A \lor B) = B$
Singular value proofs
Need hint for $\lim_{x\to 0} \frac{(x+1)^\frac{1}{x}-e}{x}$
Geodesics on the torus
A counterexample that marginal convergence in law does not imply joint convergence in law
Why this polynomial is irreducible?
When you randomly shuffle a deck of cards, what is the probability that it is a unique permutation never before configured?
Understanding proof of completeness of $L^{\infty}$
Is there a systematic way of finding the conjugacy class and/or centralizer of an element?

We counteract this answer by adding the *rigidity assumption*: Is there still a counterexample?

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$^4$, boundaryless$^2$, unbounded, uniform$^5$, and it is the $n$-skeleton of $X_{n+1}$, which is n-connected. Moreover, the distances $d_{n}$ , $d_{n+1}$ generate the same topology on $X_{n}$ and $\forall x,y \in X_{n} \ d_{n+1}(x,y) \le d_{n}(x,y)$.

Finally $(X_{n},d_{n})$ is quasi-isometric to $(X_{n+1},d_{n+1})$, through the inclusion map $X_{n} \subset X_{n+1}$, and a distance $d$ on $ \bigcup{X_{n}}$ is defined (for $x, y \in X_{n_0}$) by $d(x,y) := lim_{n (\ge n_0) \to \infty} d_{n}(x,y)$.

**Rigidity assumption**: if $S$ is a connected subspace of $X_{n}$ such that $S$ contains the geodesic paths between all its points, for $d_{n}$ **and** $d_{n+1}$, then $d_{n} = d_{n+1}$ on $S$.

- What is combinatorial homotopy theory?
- On continuity of roots of a polynomial depending on a real parameter
- Uncountable limit point of uncountable Set (Munkres Topology)
- let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?
- General relationship between braid groups and mapping class groups
- Are locally contractible spaces hereditarily paracompact?

*Definition* : Let $X:=\overline{\bigcup{X_{n}}}$ be the completion of the metric space $\bigcup{X_{n}}$ with $d$.

**Question** : Is $X$ weakly contractible ?

*Remark:* Some of these conditions could be useless for a proof, and others, highly generalized.

*Motivation*: See here for applications to *geometric group theory* and *noncommutative geometry*.

$^1$Regular (for a CW complex) : the attaching maps are homeomorphism (see this post).

$^2$Boundaryless (for a regular CW complex) : the boundary of each closed cell is contained is the union of the boundaries of other closed cells.

$^3$Constant local dimension : the topological dimension of all neighborhood of all point, is constant.

$^4$Finite type : finitely many $r$-cells ending in a fixed $(r-1)$-cell.

$^5$Uniform : For all $r$-cell $c_{1}$ and $c_{2}$, there is a neighborhood $n_{1}$ of $c_{1}$ and $n_{2}$ of $c_{1}$, such that $n_{1}$ is homeomorphic to $n_{2}$.

- Are the rationals a closed or open set in $\mathbb{R}$?
- Equivalence of continuity definitions
- The Alaoglu's Theorem
- Prob. 10 after Sec. 16 in Munkres' TOPOLOGY, 2nd edition: Which of these topologies is finer than which?
- $f$ continuous iff $\operatorname{graph}(f)$ is compact
- Tychonoff and compactness (logic) and another small logic question
- Non-normal covering of a Klein bottle by torus.
- Homeomorphism that maps a closed set to an open set?
- For $X$ contractible, deformation retract of $CX$ onto $X$.
- Given a group $ G $, how many topological/Lie group structures does $ G $ have?

- Local diffeomorphism is diffeomorphism provided one-to-one.
- Proof of $f = g \in L^1_{loc}$ if $f$ and $g$ act equally on $C_c^\infty$
- Looking for an example of a rationally indifferent cycle.
- Simple examples of $3 \times 3$ rotation matrices
- Poincare-Bendixson Theorem
- Proving ${p-1 \choose k}\equiv (-1)^{k}\pmod{p}: p \in \mathbb{P}$
- Solving a literal equation containing fractions.
- Do all square matrices have eigenvectors?
- Counting $x$ where $an < x \le (an+n)$ and lpf($x$) $ \ge \frac{n}{4}$ and $1 \le a \le n$
- How much a càdlàg (i.e., right-continuous with left limits) function can jump?
- Euler's Totient function $\forall n\ge3$, if $(\frac{\varphi(n)}{2}+1)\ \mid\ n\ $ then $\frac{\varphi(n)}{2}+1$ is prime
- Finite State Markov Chain Stationary Distribution
- sketch set satisfying $|z-2|+|z+2|\le5$
- Find min natural number $n$ so that $2^{2002}$ divides $2001^{n}-1$
- Is it wrong to say $ \sqrt{x} \times \sqrt{x} =\pm x,\forall x \in \mathbb{R}$?