= Continuous maps from products of cofinite spaces to Hausdorff spaces are constant
A <continuous map> from a nonempty finite <product space> of infinite <sets> with their <cofinite topologies> to a <Hausdorff space> is a <constant function>. Fixing all but one coordinate gives a <continuous map> from an infinite <cofinite topology>, so changing that coordinate does not change the value. Any two points in a finite <Cartesian product> can be joined by finitely many one-coordinate changes. This proves the assertion. In particular every real-valued <continuous function> on $Z\times Z$ is constant when $Z$ is the <real line> equipped with the <cofinite topology>.
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