On the finiteness of certain Zariski closed subsets of the prime spectrum of commutative Noetherian local rings

Question

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S,T$ be Zariski closed subsets of $\text{Spec}(R)$ such that if $\mathfrak p\in S, \mathfrak q \in \text{Spec}(R)$ and $\mathfrak p \subsetneq \mathfrak q$, then $\mathfrak q\in T$. If we also have $T\subseteq S$ and $T$ is a finite set, then is $S$ a finite set ?