Let $X$ be an infinite dimensional topological space such that : 


$ \forall n \in \mathbb{N}$, $ \exists X\_{n} \subset X$, $n$-dimensional subspaces verifying : 


* $\forall r<n$, the homotopy groups $\pi\_{r}(X\_{n})$ are trivial.
* $X\_{n} \subset X\_{n+1}$
* $\bigcup\_{n \in \mathbb{N}} X\_{n}$ is dense in $X$.



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



Homotopy problem for infinite dimensional topological space
Of course not. Let $X = \mathbf R^\infty \times S^1$, and let $X\_n = \mathbf R^{n-1} \times I\_n$ where $\{I\_n\}$ is an increasing sequence of intervals in $S^1$ whose union is $S^1 \setminus \{\mathrm{point}\}$.


