Defining Product Topology I have recently being studying product topology. Almost every book that I referred (including some online notes) state that we have "difficulty" in defining the product topology on, say $X \times Y$ as the Cartesian product of open sets in each of the set $X$ and $Y$.
I do not get where the "difficulty" arises.
Also, for an arbitrary product, some books say "We want the projection maps to be continuous" and then start defining the basis of the topology. Why is it so?
Can we have examples where we try to make topology by simply taking Cartesian product of the open sets in each topology and then either get a problem in the topology axioms or continuity of the projection map?

The difficulty is for infinite products.  
The product topology is defined as the smallest topology that makes the projections continuous.
It has the nice property that a product of compact spaces is compact.  
The box topology, the open sets are the all the products of open sets of the factors, is not the smallest topology for the product and though the projections are continuous, a product of compact spaces is not compact.  
Although there are yet additional ways of defining the topology of a product, they are equivalent for finite products.  
