Let $(X\_{n},d\_{n})\_{n \in \mathbb{N}}$ be a sequence of complete [geodesic metric spaces](http://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry#G) such that: 


$X\_{n}$ is a regular$^1$ [CW-complex](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/Quasi-isometry) 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 $d(x,y) := lim\_{n (\ge n\_0) \to \infty} d\_{n}(x,y)$. 


*Definition* : Let $X:=\overline{\bigcup{X\_{n}}}$ be the [completion](http://en.wikipedia.org/wiki/Complete_space#Completion) of the metric space $\bigcup{X\_{n}}$ with $d$.  

**Question** : Is $X$ [weakly contractible](http://en.wikipedia.org/wiki/Weakly_contractible) ?


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

*Motivation*: See [here](https://math.stackexchange.com/questions/478588/a-combinatorial-total-space-for-torsion-free-finitely-generated-groups) for applications to *geometric group theory* and *noncommutative geometry*.




---


$^1$Regular (for a CW complex) : the attaching maps are homeomorphism (see [this post](https://math.stackexchange.com/questions/479026/is-a-cw-complex-homeomorphic-to-a-regular-cw-complex)).  

$^2$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 $n\_{2}$.



A problem on infinite dimensional metric space
This is not true even in finite dimensions. There exists a decreasing sequence of complete Riemannian metrics on the plane, pairwise Lipschitz equivalent, such that the pointwise limit is isometric to the standard sphere without one point. Then the completion is the sphere.


To construct such a sequence, consider the metric of the punctured sphere in geodesic polar coordinates: $ds^2= dr^2+\sin^2 r\,d\varphi^2$ and add a term like $2^{-n}f(r)dr^2$, where $f(r)=1/r$ for $r$ near 0. This makes the distance to the origin infinite, so the metric is complete. But the additional term goes to zero as $n\to\infty$, so the limit is the standard metric of the punctured sphere.


To make an infinite-dimensional example, take a metric product with your favorite contractible infinite-dimensional cell complex.


