My friend and I are still stuck on:
For each , let be the set , and let be the discrete topology on . For each of the following subsets of , say whether it is open or closed (or neither or both) in the product topology.
If it helps you can think of as . We define to be the set of all functions that satisfies .
There's a nice graphical representation of the product topology on (i.e. the product of the space times). Namely, if we draw as an " axis" and as a " axis", then elements in are "graphs of functions" in the "plane". An open nbhd of an element is the set of all functions whose graphs are close to the graph of at finitely points. We get different nbhds by varying the closeness to and/or the set of finite points.
In our case the product space is , whose "plane" looks like two copies of the naturals . In other words, if you were to imagine this as a 'subset' of , it's just the set .
So remember that open sets in the infinite product topology is really just having all but finitely many the whole space and the rest are open. Since the individual factors are discrete, you only need to check that all but finitely many are the whole space.
e.g. in (a) the 10th coordinate has a specific value, but all other coordinates can be whatever, so this is certain open.