Thread: product topologies and the concept of finer

Let X and X' denote a single set in the topologies T and T', respectively; let Y and Y' denote a single set in the topologies U and U', respectively. Assume these sets are nonempty.

(i) show that if T' T and U' U, then the product topology on X' x Y' is finer than the product topology on X x Y.

(ii) does the converse of (i) hold? And how would I justify this

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.