Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$?

Question

Is there an example of a scheme $X$, irreducible, with a dense open subset $U$ so that $\dim U < \dim X$?

What about just for topological spaces?

I know for varieties this doesn't happen (though I don't have a good feeling for why).

I found this exercise in Hartshorne:

Let $X$ be an integral scheme of finite type over a field.

e) If $U$ is a non-empty open subset of $X$, then $\dim U = \dim X$.