# Is an uncountable product of connected topological spaces connected?

Let $\{X_\alpha\}_{\alpha \in J}$ be a collection of connected topological spaces, where the index set $J$ is uncountable. How can we determine whether the cartesian product of these spaces is connected or not in the product topology or in the box topology?

• Maybe you can use that $X$ is connected if and only if every continuous function $X \to \{0,1\}$ is constant. – Damien L Feb 10 '13 at 9:29
• While I think I have come across this assertion somewhere, I would appreciate if you could please give a proof thereof too. And also how to answer my query using this assertion. – Saaqib Mahmood Feb 10 '13 at 9:34
• If $f:X\to\{0,1\}$ is continuous and surjective, then $f^{-1}\big[\{0\}\big]$ and $f^{-1}\big[\{1\}\big]$ are disjoint, non-empty clopen sets whose union is $X$, and $X$ is not therefore connected. Conversely, if $X$ is not connected, then $X=U\cup V$ for some non-empty, disjoint, clopen sets $U,V$, and the function that sends every $x\in U$ to $0$ and every $x\in V$ to $1$ is continuous. This proves @Damien’s assertion. It’s not clear that it’s especially useful here, however. – Brian M. Scott Feb 10 '13 at 9:39
• I'm really sorry, but I just don't know how to accept an answer. My apologies! – Saaqib Mahmood Feb 10 '13 at 10:12