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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?

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Let $\{\langle X_\alpha,\mathscr{T}_\alpha\rangle: \alpha \in A\}$ be a family of topological spaces, let $$X = \prod_{\alpha\in A} X_\alpha,$$ and let $\mathscr{T}$ be the product topology on $X$. Let $\mathscr{F} = \{X\setminus V:V \in \mathscr{T}\}$, the family of closed sets in $\langle X,\mathscr{T}\rangle$. For each $\alpha \in A$ let $\pi_\alpha:X\to X_\alpha$ be the projection map, and let $\mathscr{F}_\alpha$ be the set of closed subsets of $X_\alpha$.

For $\alpha \in A$ let $\mathscr{S}_\alpha = \{\pi_\alpha^{-1}[U]:U \in \mathscr{T}_\alpha\}$, let $\mathscr{C}_\alpha = \{\pi_\alpha^{-1}[F]:F \in \mathscr{F}_\alpha\} = \{X\setminus V:V\in\mathscr{S}_\alpha\}$, and let $$\mathscr{S} = \bigcup_{\alpha\in A}\mathscr{S}_\alpha\text{ and }\mathscr{C} = \bigcup_{\alpha\in A}\mathscr{C}_\alpha.$$ $\mathscr{S}$ is a subbase for $\mathscr{T}$, so $\mathscr{C}$ is a closed subbase for $\mathscr{F}$. That is, every $V\in \mathscr{T}$ is a union of sets that are intersections of finitely many members of $\mathscr{S}$, so every $F\in \mathscr{F}$ is (by De Morgan’s laws) an intersection of sets that are unions of finitely many members of $\mathscr{C}$.

A union of finitely many members of $\mathscr{C}$ has the form $$\bigcup_{\alpha\in \Phi} \pi_\alpha^{-1}[F_\alpha],$$ where $\Phi$ is a finite subset of $A$, and $F_\alpha\in\mathscr{F}_\alpha$ for each $\alpha \in \Phi$. Thus, an arbitrary closed set in $X$ can be written in the form $$\bigcap_{\xi\in\Xi}\ \bigcup_{\alpha\in\Phi_\xi}\pi_\alpha^{-1}F_{\xi,\alpha},$$ where $\Xi$ is some index set, $\Phi_xi$ is a finite subset of $A$ for each $\xi\in\Xi$, and $F_{\xi,\alpha}\in\mathscr{F}_\alpha$ for each $\xi\in\Xi$ and $\alpha\in\Phi_\xi$.

Unless you limit yourself to products of finitely many spaces, you’ll have a hard time finding a simpler general form for the closed sets.

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