Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete [geodesic metric spaces][2] verifying :
   
 - $X_{n}$ is a regular [CW complex][3] of *constant local dimension* $n$.
 - $X_{n}$ is of finite type, boundaryless, unbounded and uniform. 
 - $X_{n+1}$ is [n-connected][4].
 - $X_{n}$ is the $n$-skeleton of $X_{n+1}$
 - The distance $d_{n}$ and $d_{n+1}$ generate the same topology on $X_{n}$. 
 - $\forall x,y \in X_{n}$ : $d_{n+1}(x,y) \le d_{n}(x,y)$.
 - $(X_{n},d_{n})$ is [quasi-isometric][5] to $(X_{n+1},d_{n+1})$, established by the inclusion map $X_{n} \subset X_{n+1}$.
 - Let $d$ on $ \bigcup{X_{n}}$ defined by  $d(x,y) = lim_{n \to \infty} d_{n}(x,y)$, then $d$ is a distance.    
 
**Remark** : There is a small abuse in the previous definition because $d_{n}(x,y)$ is defined only for $x, y \in X_{n}$. But because we take $n \to \infty$, there is no problem.


**Definition** : Let $X:=\overline{\bigcup{X_{n}}}$ be the [completion][6] of the metric space $\bigcup{X_{n}}$ with $d$.

> **Problem** : Is $X$ [weakly contractible][7] ?

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

____  

**Some definitions** :

**Regular** (for a CW complex) : the attaching maps are homeomorphism (see [this post][8]).   

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

**Constant local dimension** :  the topological dimension of all neighborhood of all point, is constant.

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

**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 $n_{2}$.

 


  [1]: http://math.stackexchange.com/questions/478585/homotopy-problem-for-infinite-dimensional-topological-space-iii
  [2]: http://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry#G
  [3]: http://en.wikipedia.org/wiki/CW_complex
  [4]: http://en.wikipedia.org/wiki/N-connected
  [5]: http://en.wikipedia.org/wiki/Quasi-isometry
  [6]: http://en.wikipedia.org/wiki/Complete_space#Completion
  [7]: http://en.wikipedia.org/wiki/Weakly_contractible
  [8]: http://math.stackexchange.com/questions/479026/is-a-cw-complex-homeomorphic-to-a-regular-cw-complex