Let $(X, \mathcal{U})$ be a compact uniform space which is not metrizable and and $\{U_i\}_{i=0}^{\infty}$ be a countable set of $U_i\in \mathcal{U}$ with $U_{i-1}\subseteq U_i$ and $U_1\subseteq V$. It can to see that there is open set $W\in \mathcal{U}$ such that $U_i\cap W^{c}\neq \emptyset$ (See a compact uniform space which is not metrizable).
Q. What can say about non-emptyset $U_i\cap W^{c}$, Is it an infinite set?