This post here is a specification of this post.

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

- $X_{n}$ have topological dimension $n$.
- $X_{n+1}$ is n-connected.
- $X_{n} \subset X_{n+1}$
- 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 $(X_{n+1},d_{n+1})$

**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 $X:=\overline{\bigcup{X_{n}}}$, the **complete metric space** obtained as a completion of $\bigcup{X_{n}}$ for $d$.

Question: Is $X$ weakly contractible ?

**Remark :** If yes, perhaps some of these conditions are useless in the proof, and perhaps the useful conditions can be highly generalized.

limit topology? The distance $d$ induces a topology on $X$ and $\bigcup X_{n}$ is a dense subset of $X$. Does this answer your question ? – Sébastien Palcoux Aug 20 '13 at 14:22