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**sssitex** Can somebody help me solve this problem?

Let $\displaystyle (X_i,\tau_i)$ be regular for each $\displaystyle i \in I$. Then the product space $\displaystyle (X,\tau) = \prod_{i \in I} (X_i, \tau_i)$ is regular.

My biggest problem is that I don't know how closed sets look like in a product space. Is that an infinite product of closed sets? $\displaystyle \prod_{\alpha \in A}E_\alpha$ is closed for all $\displaystyle \alpha$. Or need only a few of $\displaystyle E_\alpha$ 's to be closed? What about the rest? Is it empty?