Exercise 10, § 3

Let $A$ be a non-empty subspace of a non-empty topological space $X$. Show that the topology induced on $\mathfrak{P}_0(A)$ by the topology $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) on $\mathfrak{P}_0(X)$ (§ 2, Exercise 7) is the topology $\mathfrak{T}_\Omega$ (resp. $\mathfrak{T}_\Phi$) on $\mathfrak{P}_0(A)$.