I have been searching and searching through the Munkres Topology book for examples and trying to understand how to do this problem but I have had no luck as of yet. If someone could help out it would be much appreciated!
Let $A$ be a set and ${X_α},α∈A$ a family of topological spaces. Denote by $X$ the product $α∈A$, $X_α$.
Show that the product topology on $X$ is contained in the box topology on $X$.
Show that if $A$ is finite, then the product and box topologies are equal.
Show that for infinite $A$, if $X_α = X_β$ for all $α, β ∈ A$, then the diagonal map $\Delta: X_α \to X$ is not necessarily continuous with respect to the box topology.