Problem with the definition of the product topology? Def: The product topology is generated by sets of the form $\prod\limits_{n\in\mathbb{N}} U_n$ where each $U_n$ is open in $X_n$ and, for all but finitely many $n$, we have $U_n = X_n$.
I am slightly confused here; does it mean that all the spaces have to be the same. I mean, if a open set in $U_3$ is to be considered for inclusion in   $\prod U_n$ ,  $U_3$ is open in $X_3 $, and ....... $U_3$ is $X_3$ for all but finitely many n. It dont understand that last bit. I doesnt seem to make sense. How can $U_3$ and $X_3$ be rerelevant for $U_{100}$ and $X_{100}$?
 A: The meaning of 


*

*for all but finitely many, we have $U_n = X_n$


is 


*

*there exists a finite subset $A \subset \mathbb{N}$ such that if $n \in \mathbb{N}-A$ then $U_n=X_n$.

A: There are different kinds of product topologies.
For a family $\{X_i:i\in F\}$ of spaces:
(1).  The box product topology   on  $\prod_{i\in F}X_i$ is generated by the base of all sets of the form $\prod_{i\in F}U_i,$ where each $U_i$ is open in $X_i.$ 
(2).  The Tychonoff product topology on $\prod_{i\in F}X_i$ is generated by the base of all sets of the form $\prod_{i\in F}V_i$ where (i) each $V_i$ is open in $X_i,$ and (ii) the set $\{i\in F: V_i\ne X_i\}$ is finite. 
The Tychonoff product topology is also called the topology of point-wise convergence. It has been a very useful tool. The box product topology is stronger, usually much stronger. If $F$ is a finite set these two topologies are equal.
(3). For some classes of spaces, e.g.metric spaces, there is a uniform product topology that is weaker than the box  but stronger than the Tychonoff.
A: You have a sequence of open sets $U_1,U_2,U_3,...$; that's it, where $U_1$ is a open set of $X_1$, $U_2$ is an open set of $X_2$ and so on. i.e. $U_n$ is an open set of $X_n$ for each $n$.
And it happens that only finitely many of them are proper. For instance, suppose that only $U_1$ is strictly included in $X_1$ and $U_2$ is strictly included in $X_2$, but $U_3=X_3$, $U_4=X_4$ and so on.
