# Non-empty closed subsets with empty special fiber

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.

$$R=\mathbb{Z}_p, U=\mathrm{Spec}\:\mathbb{Z}_p\times \mathbb{Q}_p, Z=\mathrm{Spec}\:\mathbb{Q}_p\neq \emptyset$$