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**[SOLVED] Hausdorffness of the product topology**

Is it me, or is the product of an

**infinite**number of Hausdorff spaces never Hausdorff?

Recall that the product topology on

[tex]\Pi_{i\in I}X_i[/tex]

has for a basis the products of open sets

[tex]\Pi_{i\in I}O_i[/tex]

where all but

*finitely*many of those O_i are not the whole X_i.

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say I is countable for simplicity and consider x=(x1,...) and y=(y1,...) two distinct points in the product space. I don't see how we can find two ngbh of x and y that do not intersect!