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-TsenCommented Jan 2, 2019 at 18:09
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$\begingroup$ @Gro-Tsen As far as I understand, your link is more about point set topology aspects of box topology... $\endgroup$– OfraCommented Jan 3, 2019 at 0:20
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$\begingroup$ what do you mean by "weakly equivalent to a point"? $\endgroup$– Henno BrandsmaCommented Jan 3, 2019 at 5:33
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1$\begingroup$ 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 GunninghamCommented Jan 3, 2019 at 11:56
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$\begingroup$ But are there counterexamples (to being null-homotopic) when connectedness is preserved? $\endgroup$– Chris GerigCommented Jan 3, 2019 at 15:52
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