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Let $\{(X_\alpha,\mathscr{T}_\alpha):\alpha\in\Lambda\}$ be an indexed family of topological spaces, and for each $\alpha\in\Lambda$ let $A_\alpha\subseteq{X_\alpha}$, then $\overline{\prod_{\alpha\in\Lambda}A_\alpha}=\prod_{\alpha\in\Lambda}\overline{A_\alpha}$.

I have tried to prove this and I have gotten the expression $\overline{\prod_{\alpha\in\Lambda}\overline{A_\alpha}}=\prod_{\alpha\in\Lambda}\overline{A_\alpha}$, because if I choose a subset $A_\beta$ of $X_\beta$, i know that $\prod_{\alpha\in\Lambda}\overline{A_\alpha}$ (where $A_\alpha=X_\alpha$ if $\alpha\neq{\beta})$ is a closed subset of $\prod_{\alpha\in\Lambda}X_\alpha$.

  • I need your help to end the proof.
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