i)

The product topology on X x Y consists of the sets, which are unions of sets of the form M x N , with M in T and N in U , because the sets of the form MxN are a basis for the product topology.

Since every topology is closed under union, it is enough to show that MxN is an element of the product topology on X' x Y', which is clear since T' is finer than T and U' finer than U, i.e. M is in T' and N is in U' and so MxN is in the product topology of T' and U' as well.