I didn't exactly know if I should publish this in Law SE or here, but since this can also lead to a politics discussion, I'll stay here. By the way this is not in any way my situation, just a doubt that came to my mind.
The EU is not a sovereign country per se, but it is fairly developed as a sui generis entity.
In a typical country, you usually have a way to apply to become a full fledged citizen after you have accomplished some requisites. What about the EU? Imagine a UK citizen, who would like to stay as an EU citizen, is that even possible or are you restricted to your country's decision? What about prominent/rich people? (Becoming a citizen is fairly easy in most countries when you bring money with you)