How would I go about showing that a Tychonoff space $(T_{3.5})$ is finitely productive; or is there a counter-example disproving it? That is, if $A$ and $B$ are Tychonoff, then $A \times B$ is Tychonoff. In this context Tychonoff is defined as a space that is $T_1$ and where for any point $x$ and closed set $C$ such that $x \notin C$, there exists a continuous function with range $[0,1]$ such that $f(x) = 0$, and $f(C) = 1$.
I understand how to show that a regular $T_3$ space is finitely productive, however I have no idea how to show it for $T_{3.5}$.