Let $X$ be a compact Hausdorff space and $\mathcal{V}$ be a finite open cover of $X$. Since $X$ is a compact Hausdorff space, hence it has a uniformity $\mathcal{U}$. Is there an entourage $D$ in uniformity $\mathcal{U}$, such that if $(x, y)\in D$, then there is open set $V$ in finite open cover $\mathcal{V}$, $V\in\mathcal{V}$, such that $\{x, y\}\subseteq V$.
Would you please help me to know it?
For every $x \in X$ we find some $V_x \in \mathcal{V}$ such that $x \in V_x$.
We can find an entourage $U^x_1 \in \mathcal{U}$ such that $U^x_1[x] \subseteq V_x$, because $\mathcal{U}$ is the uniformity for this topology.
By standard axioms we can find a symmetric entourage $U^x_2 \in \mathcal{U}$ such that $U^x_2 \circ U^x_2 \subseteq U^x_1$.
Then $\{U^x_2[x]\mid x \in X\}$ is a cover of $X$ that has a finite subcover, say indexed by $x_1,\ldots, x_n$.
Then $U=\bigcap_{i=1}^n U^{x_i}_2$ is also in $\mathcal{U}$ and has the required property that $(x,y) \in U$ implies there is some $V \in \mathcal{V}$ that contains $\{x,y\}$.