# A product topology where we allow countably many open sets

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?

• 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. – Hanul Jeon Oct 5 '17 at 5:51
• @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. – Santana Afton Oct 5 '17 at 11:43
• @HanulJeon indeed, its not a $\Sigma$-product. – Henno Brandsma Oct 6 '17 at 5:57
• This paper may help. – user90189 Oct 12 '17 at 0:13
• @ChrisCulter: only if the $X_\alpha$ already have a $G_\delta$ topology ("P spaces") – Dap Oct 15 '17 at 22:10