How is it possible for a countably infinite product space to be discrete?
If we let each $(X_i,T_i)$ be topological spaces with more than $1$ point and $$\prod^{\infty}_{i=1}(X_i,T_i)$$ be the countably infinite product space, then $$\prod^{\infty}_{i=1}\{a_i\} \subseteq \prod^{\infty}_{i=1}(X_i,T_i)$$ and is therefore an open set in the product space. But in a product space, open sets are of the form $U = \{\prod ^{\infty}_{i=1}O_i : O_i \in T_i$ and $O_i = X_i$ for all but a finite number of $i \}$.
This seems to be a contradiction. Can someone explain to me what I'm not understanding?