5
$\begingroup$

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 ?

Improve this question
$\endgroup$
5
  • $\begingroup$ Related question: mathoverflow.net/questions/209661/… $\endgroup$
    – Gro-Tsen
    Jan 2, 2019 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, 2019 at 0:20
  • $\begingroup$ what do you mean by "weakly equivalent to a point"? $\endgroup$
    – Henno Brandsma
    Jan 3, 2019 at 5:33
  • 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 Gunningham
    Jan 3, 2019 at 11:56
  • $\begingroup$ But are there counterexamples (to being null-homotopic) when connectedness is preserved? $\endgroup$
    – Chris Gerig
    Jan 3, 2019 at 15:52

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.