Given a dcpo $\mathcal{X} = (\le, X)$, consider the set $\mathcal{X}^{sub}$ of all sub-dcpos of $\mathcal{X}$. Can one define a partial order $\le_{sub}$ on $\mathcal{X}^{sub}$ such that $( \le_{sub}, \mathcal{X}^{sub})$ is a dcpo?
I supose you may add, what conditions on $\mathcal{X}$ allow us to define $\le_{sub}$ on $\mathcal{X}^{sub}$ such that $( \le_{sub}, \mathcal{X}^{sub})$ is a dcpo?