More accurately, those sets form a base for the product topology. It is not true that every open set is of that form. For a set U to be open in the product topology, it is necessary that every point of U should contain a basic neighbourhood that is contained in U.

For example. take set (b). Let . If then there exists m such that f(m)=0. Then the set is an open neighbourhood of f contained in B. Therefore B is open.

It's usually more difficult to check when a set is closed. You have to look at its complement and decide whether that is open. Sometimes this is straightforward. For example, the complement of set (a) is the set of all f such that f(10)=1. That is open, so set (a) is closed as well as open.

For a slightly less easy example, look at set (c). Let . If then there exists m such that f(m)=f(m+1)=0. Then is an open neighbourhood of f containing no points of C. Therefore the complement of C is open and so C is closed.

I think that illustrates the main techniques you will need to answer the rest of the question.