I am reading Miln'es Etale Cohomology (1980 Princeton University Press). In page 46 to give some exampeles of $E$-morphisms he writes $E=(et)$ of all etale morphisms of finite-type. Next page he writes that $E$-morphisms are open and any open immersion is an $E$-morphism.
Thus if I understand correctly, given any open immersion $U\to X$ it has to be an $E$-morphism. In this case $E$ is etale of finite-type. Therefore any open immersions needs to be of finite-type. But that is false.
Could you please tell me if my way of thinking is flawed?