Is an affine open subset of a scheme stable under specialisations?

Question

In Stacks project tag 05JL, we find the statement that $f(X)\cap U$ is stable under specialisations when $f(X)$ is, where $U$ is an affine open of $U$. So my question is:

Is an affine open subset of a scheme stable under specialisations?

I cannot find a reason for this to be true: maybe another affine open subset $V$ intersects with $U$ and a point in $U$ specialises to $V$?

Thanks in advance for any help or reference.

Answer 1

Now I understand the statement in question:

$f(X)\cap U$ is stable under specialisations in $U$ whenever $f(X)$ is so in $X$.

Namely, $U$ need not be stable under specialisations for this statement to be true, though I cannot find a counter-example.