Let $R$ be a dvr and $U\to \text{Spec}(R)$ an affine smooth $R$-scheme with non-empty special fiber $U\_0$.


Let $Z\subset U$ be a closed subset. Assume the intersection of $Z$ with $U\_0$ is empty.



> 
> Is $Z$ empty?
> 
> 
> 


If $U\to \text{Spec}(R)$ was proper then the answer would be yes because the image of $Z$ could only be the closed point in $\text{Spec}(R)$.


A closed point of $Z$ could map to the generic point of $\text{Spec}(R)$ without properness of $U\to \text{Spec}(R)$, so $Z$ could be contained in the generic fiber, so the answer should be no.


I'm just having a bit of trouble picturing the situation visually.



Non-empty closed subsets with empty special fiber
$R=\mathbb{Z}\_p, U=\mathrm{Spec}\:\mathbb{Z}\_p\times \mathbb{Q}\_p, Z=\mathrm{Spec}\:\mathbb{Q}\_p\neq \emptyset$


