(Comparison of the box and product topologies). The box topology on $\prod X_\alpha$ has as basis all sets of the form $\prod U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for each $\alpha$. The product topology on $\prod X_\alpha$ has as basis all sets of the form $\prod > U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for each $\alpha$ and $U_\alpha$ equals $X_\alpha$ except for finitely many values of $\alpha$.
Munkres' Topology doesn't provide some examples to illustrate the difference between box topology and product topology, so can someone give some examples to help us understand these two concepts? Is $\prod X_\alpha$ a basis element of the product topology?
$\prod_\alpha X_\alpha$ is certainly open, as all but zero (zero is finite) sets in the product equal $X_\alpha$.