Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :

• $X_{n}$ is a regular CW complex of constant local dimension $n$.
• $X_{n}$ is of finite type, boundaryless, unbounded and uniform.
• $X_{n+1}$ is n-connected.
• $X_{n}$ 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 to $(X_{n+1},d_{n+1})$, established by the inclusion map $X_{n} \subset X_{n+1}$.

Definition : Let $d$ be a distance on $ \bigcup{X_{n}}$, defined as follows : $d(x,y) = lim_{n \to \infty} d_{n}(x,y)$.

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 of the metric space $\bigcup{X_{n}}$ with $d$.

Problem : Is $X$ weakly contractible ?

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).

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 $n_{2}$.