It is not a typo and for example arises when testing composite hypotheses, such as a one-sided test of the mean of a distribution, as in $H_0:\mu\leq0$ against $H_0:\mu>0$. We choose the critical value (equivalently, $\omega$) so that the probability of a type-I error is still just $\alpha$ even when $\mu$ is very close to being in the set specified by $H_1$, i.e., when $\mu=0$, i.e., when the probability of a type-I error is highest.
Now if, say, the true $\mu$ equals $-5$, the probability of obtaining a sample such that $t\in\omega$ will be much smaller. For example, suppose we sample from a $N(\mu,1)$ distribution. Then, our t-statistic for known $\sigma$ and testing against positive $\mu$ would simply be $t=\sqrt{n}\bar{x}$ and follow a $N(\mu,1)$ distribution, too.
The plot below shows the probability of rejecting $H_0$, i.e., the power function. The region to the left of the vertical dashes is where the null is true, and you can see that the rejection probability is .05 only at $\mu=0$.

mu <- seq(-3,3,by=.02)
plot(mu,1-pnorm(1.645,mean=mu),type="l",lwd=2,col="sienna3")
abline(h=.05,lty=2)
abline(v=.0,lty=2)