This question comes from Proposition 2.6 in Chapter 2 of Hartshorne's *Algebraic Geometry*. In my edition, that's on page 78.

For a variety $V$, Hartshorne defines the topological space $t(V)$ to consist of the nonempty closed irreducible subsets of $V$, where the closed sets of $t(V)$ are of the form $t(Y)$ for $Y$ closed in $V$. He then defines a map $\alpha: V \rightarrow t(V)$ where P gets sent to {P} in $t(V)$. The claim is that $(t(V), \alpha_*(\mathcal{O}_V))$ is a scheme. I understand why this is true if $V$ is affine, but I have been unable to show $(t(V), \alpha_*(\mathcal{O}_V))$ is a scheme for an arbitrary variety $V$.

I had hoped to show that if $U$ is an affine open subset of $V$, then $t(U)$ is isomorphic to an open subset of $t(V)$. I used the map from $t(U)$ into $t(V)$ where we send an irreducible subset $W$ in $U$ to the smallest irreducible subset of $V$ containing $W$. However, although the image of of $t(U)$ is contained in $[t(U^c)]^c$, I don't believe these are equal.