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 ?