homotopy type of box topology.

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 ?

• Related question: mathoverflow.net/questions/209661/… – Gro-Tsen Jan 2 at 18:09
• @Gro-Tsen As far as I understand, your link is more about point set topology aspects of box topology... – Ofra Jan 3 at 0:20
• what do you mean by "weakly equivalent to a point"? – Henno Brandsma Jan 3 at 5:33
• 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. – Sam Gunningham Jan 3 at 11:56
• But are there counterexamples (to being null-homotopic) when connectedness is preserved? – Chris Gerig Jan 3 at 15:52