Why in uniform space contains small sets should be nonempty meets?

Question

In the figure, $\delta(F)<V$ means $F\times F\subseteq V$. Wy $\bigcap\mathcal F\neq\varnothing$? If we let $\mathcal F=\{\{x\},\{y\}\}$, then it clearly satisfies the condition: contains arbitrary small sets, but $\bigcap\mathcal F=\varnothing$ .