Neighborhood of $\Delta_X$ in $X\times X$

Question

Let $X$ be a first countable, locally compact, paracompact, Hausdorff space. Let $\Delta_X=\{(x, x)| x\in X\}$ and let $U^{2}=\{(x, y)| \exists z_0=x, z_1, z_2=y ; (x, z_1), (z_1, y)\in U\}$

Question. For open set $D$ of $\Delta_X$, is there an open set $U$ of $\Delta_X$ with $U^{2}\subseteq D$?

What is the relation between $U^{2}$ and $U$?

Answer 1

If $D$ is an open neighbourhood of $\Delta_X$, $\mathcal{U} = \{D[x] :x \in X\}$ is an open cover of $X$. By paracompactness, this has an open star-refinement $\mathcal{V}$. Define a neighbourhood $U$ of $\Delta_X$ from $\mathcal{V}$ in the standard way: $U = \cup\{V \times V: V \in \mathcal{V}\}$.

I think this should work. ( I only used paracompactness in its star refinement form). See theorem 2 in my note.