This is pretty basic, but I just learned about the product topology in class, and it seems to me that it doesn't agree with how it is defined – obviously, I'm wrong, but I was hoping someone could explain to me exactly where my thoughts are going wrong.
Working in the simple case of two sets for illustrative purposes: Let $(X,T_x)$ and $(Y,T_y)$ be topological spaces. Then we define the product topology on $X \times Y$ to be $T_{X \times Y} = \{U \times V: U \in T_x, V \in T_y\}$. This is supposed to be the weakest topology on $X \times Y$ such that the projection operator $p_i$ is continuous for all $i$. However, let us take the case $P \times Q \subset X \times Y$ such that $P \in T_x$ but $Q \notin T_y$. By the definition of product topology this isn't an open set. Then, $p_1[P \times Q] = P \in T_x$. Since $P$ is an open set, but $p_1^{-1}[P]$ isn't an open set, wouldn't this mean that $p_1$ is not continous, in contradition to the definition of the product topology?
Best Answer
No, we do not define the product topology to be $\{U\times V:U\in T_X,V\in T_Y\}$ we define it to the the topology with basis $\{U\times V:U\in T_X,V\in T_Y\}$.
In all cases, if $U\in T_X$ then $p^{-1}(U)=U\times Y$ which is open in $X\times Y$.