Let $\{X_{\alpha}\}$ be an uncountable collection of topological spaces indexed by the set $J$. For the space $\prod_{\alpha}X_{\alpha}$, consider the topology $\tau$ generated by the basis

$$\mathcal{B}=\left\{\left(\prod_{\alpha\in S}U_{\alpha}\right)\times\left(\prod_{\beta\notin S}X_{\beta}\right): U_{\alpha}\text{ open in }X_{\alpha},S\subseteq J\right\}.$$

If $S$ is arbitrary, then $\tau$ is the box topology. If $S$ is finite, then $\tau$ is the product topology. What if we allow $S$ to be countable? Has this topology been studied at all? Does it have a use somewhere?

  • $\begingroup$ I've search related terms in Google. If my searching is correct, your concept is called the $\Sigma$-product. A blog of Dan Ma introduces many facts about the $\Sigma$-product. $\endgroup$ – Hanul Jeon Oct 5 '17 at 5:51
  • $\begingroup$ @HanulJeon Unless I'm mistaken, it looks like Dan Ma defines a $\Sigma$-product as a particular subspace of $X= \prod X_{\alpha}$ equipped with the subspace topology, not giving $X$ itself a separate topology. $\endgroup$ – Santana Afton Oct 5 '17 at 11:43
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    $\begingroup$ @HanulJeon indeed, its not a $\Sigma$-product. $\endgroup$ – Henno Brandsma Oct 6 '17 at 5:57
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    $\begingroup$ This paper may help. $\endgroup$ – user90189 Oct 12 '17 at 0:13
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    $\begingroup$ @ChrisCulter: only if the $X_\alpha$ already have a $G_\delta$ topology ("P spaces") $\endgroup$ – Dap Oct 15 '17 at 22:10

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