Let D be an integral domain, Spec(D) the set of prime ideals of D, and X a subspace of the Zariski topology on Spec(D). We show that X is compact if and only if given any ideal I of D with $I{ subseteq}P$ for all $P{in}X$, there exists a finitely generated idea $J{subseteq}I$ such that $J{ subseteq}P$ for all $P{in}X$. We also prove that if D = ${cap}_{P{in}X}D_P$ and if * is the star-operation on D induced by X, then X is compact if and only if * $_f$-Max(D) ${subseteq}$X. As a corollary, we have that t-Max(D) is compact and that ${mathcal{P}}$(D) = {P${in}$ Spec(D)$|$P is minimal over (a : b) for some a, b${in}$D} is compact if and only if t-Max(D) ${subseteq};{mathcal{P}}$(D).