Vtyjkyuk

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 $T_3.5$.

You can find a proof in any book on general topology. In fact, the following is true: Let $X_alpha$, $alpha in A$, be a family of Tychonoff spaces. Then $P = Pi_alpha in A X_alpha$ is a Tychonoff space.

We have to show that each $p in P$ and each open neighborhood $U$ of $p$ in $P$ admits a continuous $f : P to I$ such that $f(p) = 1$ and $f(x) = 0$ for $x notin U$.

Choose finitely many $alpha_i in A$ and open neigborhoods $U_i$ of $p_alpha_i$ in $X_alpha_i$ such that $bigcap_i pi^-1_alpha_i(U_i) subset U$, where $pi_alpha_i : P to X_alpha_i$ denotes projection. There exist continuous $f_i : X_alpha_i to I$ such that $f_i(p_alpha_i) = 1$ and $f_i(x_alpha_i) = 0$ for $x_alpha_i notin U_alpha_i$. Define $f : P to I, f = min_i f_i circ pi_alpha_i $. Then $f(p) = 1$ and $f(x) = 0$ for $x notin bigcap_i pi^-1_alpha_i(U_i)$, in particular for $x notin U$.