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There's a theorem in Munkers that says if Xa is a Hausdorff space then infinite or finite(the book didn't mention if "a" comes from a finite or infinite set) product of them is Hausdorff with box or product topology I could prove it for box topology but it seems that this theorem works for product topology if "a" comes from a finite set because open set in product topology is product of finitely many open sets. if "a" is infinite that causes problem

Lavender
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    The goal is to separate a pair of distinct functions in the product. That means there's at least one coordinate where they differ. Try using the Hausdorff property of the corresponding coordinate to get a basic open set separating them. – Steven Clontz Mar 31 '24 at 13:01

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