Question: The short argument outlined in the following link demonstrates that a general cartesian product of Hausdorff spaces is itself Hausdorff.

<http://planetmath.org/ProductTopologyPreservesTheHausdorffProperty.html>

But this argument doesn't seem to depend on *all* of the factors of the product topology being Hausdorff (instead only one). Am I missing something, or is it enough for a **single** factor in a product topology to be Hausdorff to guarantee that the entire product topology becomes Hausdorff?

Answer: 
Read the proof again. The factor we require to be Hausdorff is one in which $x$ and $y$ are different. This could be any factor, so we need every factor to be Hausdorff for the proof to work. And indeed the result is false otherwise: consider for example the product of a Hausdorff space with a two-point trivial space.
A single Hausdorff factor does not sufffice, since **every** topological space, Hausdorff or not, is homeomorphic to the product of the space itself and the singleton **Hausdorff** space: $$X\cong X\times\{\bigstar\} $$