In Johnstone´s Stone Spaces, he introduces the concept of soberification of a topological space:
Let $X$ be a topological space and $\Omega(X)$ the lattice of open subsets of $X$, the soberification of $X$ is $pt(\Omega(X))$, where $pt(A)$ is the set of prime elements of $A$, for any set $A$.
$pt(\Omega(X))$ is the of all principal prime open sets, i.e., open sets whose complements are irreducible closed sets.
The function $\psi: X \to pt(\Omega(X))$ sends a point $x\in X$ to $\overline{ \{ x\} }^c$.
$pt(\Omega(X))$ has the topology of subspace of $Spec(\Omega(X))$.
I would like to prove that this function is always continuous, and it is injective if and only if $X$ is $T_0$.
To prove that $\psi$ is injective if and only if $X$ is $T_0$ is really simple, because $\psi(x)=\psi(y)$ iff $\overline{ \{ x\} }^c$=$\overline{ \{ y\} }^c$ iff $x, y$ are contained in exactly the same open sets. So, $\psi(x)=\psi(y)$ implies $x=y$ iff $X$ is $T_0$.
So, the proof of continuity is left. I was also wondering when this function is onto.
Any help or hint is appreciated.
Edit: So far I've managed to come up with a characterization of $X$ so that $\psi$ is onto:
$\psi$ is onto if and only if for any $x,y\in X$ such that $y\notin\overline{\{x\}}$ we have $\overline{\{x,y\}}$ is not connected.
Is there a name for this kind of spaces?