Is there a way to see intuitively the basis and subbasis of $\displaystyle\prod_{\alpha\in I}X_\alpha,\tau_{\alpha}$ ?
The basis is $\beta_{\pi}=\{\bigcap_{\alpha\in F}\prod_\alpha^{-1}(U_\alpha):U_\alpha\in\tau_\alpha,F<\infty\}$ and the subbasis is $ S_\pi=\{\prod_\alpha^{-1}(U_\alpha):U_\alpha \in \tau_\alpha,\alpha\in I)\}$
And another question: $S_\pi$ the elements of the set are like $\pi_1,\pi_2,\dots$ and therefore they can be represented as the union.
And the idea generalizes to any kind of set, if a set has the form of $S_\pi,$ then it can be represented as the union, i.e. $\bigcup S_\pi. $
Am I correct?