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?
[Math] Is an uncountable product of connected topological spaces connected
connectednessgeneral-topology
Best Answer
The box product of infinitely many non-trivial Tikhonov spaces is never connected; this is Theorem 1.3(iii) of Scott W. Williams, Box Products, in the Handbook of Set-Theoretic Topology, K. Kunen & J.E. Vaughan, eds., North-Holland, 1984. An arbitrary Tikhonov product of connected spaces, however, is always connected; you’ll find a proof here.