Understanding on the basis of topology on infinite product space For infinite product space, the product topology is generated by the open sets that place open restrictions only in finitely many coordinates. I try to understand why it must be the restrictions on finitely many coordinates, but not infinitely many. see Infinite Product Space
Take example of $\mathbb R^N$, where $N = \{1,2,\ldots\}$. What if I generate a topology with the basis $B = \{ \Pi_{i=1}^{\infty} O_i | O_i \hbox{ is an open set in }\mathbb R\}$?
 A: The topology you describe has been studied, but is less natural. It's called the box topology on the product. The standard product topology is the minimal topology that makes the projections all continuous, and has all sort of nice properties (like a map into a product is continuous iff all the compositions with the projections are continuous). Also the standard product topology has nice theorems: a countable product of metrisable spaces is metrisable, any product of compact spaces is compact, any product of connected spaces is connected etc. 
For the box topology, this all fails: no non-trivial box-topology of infinitely many spaces is connected, compact or metrisable. It's sometimes used as a counterexample, and there are some interesting questions about it, but it's less natural and has few theorems like the standard product has.
A: The topology you describe is called the box topology and is strictly finer than the product topology. If you think of the product topology as the initial topology of the projections with its universal property, you may see easier that you only need finite restrictions.
