It is well-known that when we have a spectral space $Y$ and a subset $E \subseteq Y$ that is closed with respect to the patch topology on $Y$, then $E$, with the subspace topology induced by $Y$, is itself a spectral space.
Now let $X$ be a scheme. Since $X$ carries the structure of a topological space, we can still form the patch topology on $X$. Suppose that $E \subseteq X$ is closed in the patch topology.
What can we say about the restriction of $X$ to $E$?
It is clear that whenever $E$ is an affine open subset (hence closed in the patch topology), we have an affine scheme. Also, trivially, whenever $E$ is contained in some open affine, the underlying set is a spectral space. But I fail to see much more.
