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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 ?

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What does dimension mean here? Something like Lebesgue covering dimension, or do you really only care about CW complexes or something? –  Qiaochu Yuan Aug 19 '13 at 9:38
    
@QiaochuYuan : thank you for your comment. My needs are close to CW complexes, but I ask the question in general. The Lebesgue covering dimension seems coincide with the dimension of a CW-complex. I could add the condition that every open subspaces have the same Lebesgue covering dimension (to exclude the disjoint union of $\mathbb{S}^{1}$ and $\mathbb{S}^{2}$, for example). Do you know a name for such spaces? –  Sébastien Palcoux Aug 19 '13 at 11:32
    
I have posted a specification here. –  Sébastien Palcoux Aug 19 '13 at 15:09
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2 Answers 2

up vote 6 down vote accepted

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 $S^1 \setminus \{\mathrm{point}\}$.

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oups... your $X_{n}$ is not $n$-dimensional. –  Sébastien Palcoux Aug 19 '13 at 9:11
    
But you can take $H=R^{\infty}$ and $H_n =R^{n-1}$ instead. –  Johannes Ebert Aug 19 '13 at 9:15
    
Oh, I missed the condition on the dimension of $X_n$. I modified my example by your suggestion, Johannes. –  Dan Petersen Aug 19 '13 at 9:22
    
I have posted a specification here. –  Sébastien Palcoux Aug 19 '13 at 15:10
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As Dan pointed out, this is wrong in general, but there are situations where an approximation result does hold. One example for such results is due to Palais ('On the homotopy type of certain groups of operators', Topology, Vol 3). It applies when $X$ is an open subset of a Banach space, with some extra properties.

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Thank you. You enlighten me that I need to find some extra properties. I have some concrete examples in mind. I will read your reference, and perhaps I will open a more specific post. –  Sébastien Palcoux Aug 19 '13 at 11:45
    
I have posted a specification here. –  Sébastien Palcoux Aug 19 '13 at 15:10
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