$$ f^{\ast}(S) = \\{x \in X: \; f(x) \in S\\} \subseteq \\{x \in X:\; f(x) \in S\\} = f^{\ast}(T) $$

Should this be

$$ f^{\ast}(S) = \\{x \in X: \; f(x) \in S\\} \subseteq \\{x \in X:\; f(x) \in T\\} = f^{\ast}(T) $$