Is it true that a topology space X with a zeroset diagonal is first countable? 


what if X is additionally CCC?



zeroset-diagonal
All it takes is a countable, not first-countable, Tychonoff space, say a countable dense subset, $D$, of the Cantor cube $2^{\mathfrak{c}}$. For every point $(d,e)$ off the diagonal there is a continuous function $f\_{(d,e)}$ on $D^2$ that is zero on the diagonal and has value $1$ at $(d,e)$. Now enumerate the complement of the diagonal as $\lbrace(d\_n,e\_n):n\in\mathbb{N}\rbrace$ and define $f=\sum\_{n=1}^\infty 2^{-n}f\_{(d\_n,e\_n)}$. 


