neighborhood of diagonal $X$

Question

Let $X$ be a topological space and $\Delta_X=\{(x, x): x\in X\}$. Let $D$ be a closed neighborhood of $\Delta_X$ and $D\neq \Delta_X$, is there an open neighborhood $N$ of $\Delta_X$, $N\neq \Delta_X$, such that $N\subseteq D$?

Answer 1

Yes. The interior $D^o$ would be open and contain $\Delta_x$, as well as being contained in $D$.