I have this basic question, I want to write the closed sets of a product space (under the product topology) in a general way. This problem is more related to set theory than to topology, so if someone does't knows topology but can "evalf" this expression, also can help me.
Any open set in this topology is a union of product of open sets, i.e., $$ J = \bigcup _{\alpha \in A} \prod_{\beta \in B} X_{\alpha, \beta} $$ and every closed set must be of the form $$ J^c = \bigcap_{\alpha \in A} \left( \prod_{\beta \in B} X_{\alpha,\beta} \right)^c $$ so I want to rewrite this part $$ \left( \prod_{\beta \in B} X_{\alpha, \beta} \right)^c . $$
It's easy to see that $$ (A\times B)^c = (A^c \times B^c) \cup ( A^c\times B) \cup (A\times B^c), $$ but for arbitrary products, can I do something? Or at least, can I write the closed sets of the product in a general way, just like can I with open sets?