I'm not quite sure where such a concept fits. Suppose $X$ is a topological space. I know then that the diagonal $\Delta=\{(x,x)\ | x\in X\}$, so $\Delta\subseteq X\times X$. What then would a neighborhood of the diagonal be comprised of?
By this I mean, suppose $E\in\mathscr{N}_\Delta$, where $\mathscr{N}_\Delta$ is the filter of all neighborhoods of the diagonal. Is $E\in\mathscr{P}(X)$, or is $E\in\mathscr{P}(X\times X)$? What is the criterion for a subset to be in a neighborhood of $\Delta$? My guess is that $E\in\mathscr{P}(X\times X)$, and if this is correct, we would say
$$(x,x')\in E\quad\text{if and only if...}?$$
Thanks!