I actually have two questions. One about topological comparisons between those two topologies and one about the identity map. First, I just wanted to make sure a proof I was using was correct or not... I've battled with it for awhile and I want some criticism.

The exercise was: Let be an indexed family of copies of the real line under the usual topology and let . Show that the product topology is a proper subset of the box topology on .

My proof was this:

For this proof, it will suffice to show that where and are the bases of the Box and Product topologies, respectively. We will first consider . We know, then, that but not in since every set in has for all but a finite number of . Thus, . Now we will show that . We first remember that, in , any, all or no from will be . only requires that each individual , and this is true whether or . Hence, we can conclude that any element of is in and .

The second question was this:

For each , let be the identity map . Show that each is continuous relative to the usual topology on .

I am not sure how to proceed with this. Any hints or help would be much appreciated.