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?
You are correct that there is an error here, and not every open immersion is etale of finite-type. But Milne is not saying that an open immersion has to be an $E$-morphism as a part of any definition; he is just observing that this property holds in these examples (except that he is wrong about one of them). For what it's worth, in most cases of interest the base scheme is Noetherian, and then it is true that all open immersions are finite type.