I understand that an open set is just a subset of a space that obeys the definition of a topology.
I understand that a basis is a collection of open subsets of a space such that every open set of the space is a union of members of the basis.
But I'm confused for product spaces:
Product spaces $(\prod X_i, T')$ are defined with a basis for $T'$ being generated by $\prod O_i$. So by the definition of a basis, each $\prod O_i$ should be an open set in $(\prod X_i, T')$. But to use a specific case: If $(0,1)\times(0,1)$ and $(2,3)\times(2,3)$ are open sets then their union should be an open set in $R^2$. But this defies the definition of a topology since no $O_1\times O_2$ equals this for $O_1, O_2$ being open subsets in $R$, respectively.
Can someone explain what I'm misunderstanding here?