On the Density Property of P𝒮C Fields

Let 𝒮 be a finite set of finite and real local primes of a field K. We prove two results. First, a PAC field over K is already PAC over each nonempty 𝒮‐open subset of K; if K is global, this implies that it is also PAC over each separable Hilbert subset of K. Second, each P𝒮C field inside K_tot,𝒮 has the 𝒮‐density property.