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Suppose that $X$ is weakly equivalent to a point. Let $I$ be a set. Does $\prod_{i\in I}X$ weakly equivalent to a point, where $\prod_{i\in I}X$ is equipped with box topology ?

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  • $\begingroup$ Related question: mathoverflow.net/questions/209661/… $\endgroup$ – Gro-Tsen Jan 2 at 18:09
  • $\begingroup$ @Gro-Tsen As far as I understand, your link is more about point set topology aspects of box topology... $\endgroup$ – Ofra Jan 3 at 0:20
  • $\begingroup$ what do you mean by "weakly equivalent to a point"? $\endgroup$ – Henno Brandsma Jan 3 at 5:33
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    $\begingroup$ As mentioned in the linked question in the comments above, (path) connectedness is not preserved by box product. For example, the box topology on $\mathbb{R}^\omega$ is not (path) connected. Thus the answer to your question seems to be no. $\endgroup$ – Sam Gunningham Jan 3 at 11:56
  • $\begingroup$ But are there counterexamples (to being null-homotopic) when connectedness is preserved? $\endgroup$ – Chris Gerig Jan 3 at 15:52

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