Type I and Type II errors in Hypothesis Testing

I am confused about the last highlighted sentence regarding finding a subset S, for which BOTH Type I and Type II error probabilities are 0. For $$P_\theta S$$, which is the probability with which we reject the null hypothesis $$H_0$$, if the actual $$\theta$$ says that our $$H_0$$ is true, then shouldn't the probability of rejecting $$H_0$$ be 0 and not 1? Should I be imagining $$S$$ to be $$C^c$$ from the definition of a critical region?

You are right, it looks like 0 and 1 have got accidentally changed in the last highlighted sentence. If $$P_\theta S$$ is the probability of the critical region, it should ideally be 0, when $$\theta\in \Theta_0$$ ($$H_0$$ true) and it should ideally be 1, $$\theta\in \Theta_1$$ ($$H_0$$ false). Note that the probability of type II error was $$P_\theta S^C=1-P_\theta S$$, $$\theta \in \Theta_1$$ ($$S^C$$ is the complement of $$S$$). So, now both errors would be 0.