= Hausdorff factorization through a cofinite product factor
{c}
{title2=$F:Z\times T\to Y\text{ continuous}\ \Longrightarrow\ F=g\circ\pi_T$}
Let $Z$ be a nonempty infinite <set> with the <cofinite topology>, let $T$ be any <topological space>, and let $Y$ be <Hausdorff>. For a <continuous map> $F:Z\times T\to Y$, every slice $z\mapsto F(z,t)$ is constant. Thus $F(z,t)=g(t)$, where $g(t)=F(z_0,t)$ for any fixed $z_0$. The map $g$ is <continuous>, since $t\mapsto(z_0,t)$ is <continuous> into the <product topology>. It is unique because the <coordinate projection> $\pi_T$ is a <surjection>. If $F$ is a <surjection>, so is $g$.
For $T=[0,1]$ and $Y$ <homeomorphic> to the <circle>, a surjective $g$ cannot be injective: otherwise the <compact-to-Hausdorff continuous bijection theorem> would contradict the fact that the <circle and interval are not homeomorphic>. Hence distinct $a,b\in[0,1]$ satisfy $F(z,a)=F(z,b)$ simultaneously for all $z\in Z$.
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