A totally order-disconnected space (TOD) is a tuple $(P, \leq, \tau)$ where $(P, \leq)$ is a poset and $(P,\tau)$ is a topological space such that for $x\not\leq y$ in $P$ there is a clopen down-set that contains $y$, but not $x$. If $P, Q$ are TODs then a map $f:P\to Q$ is a *morphism* if $f$ is continuous and order-preserving. The set of all morphisms from $P$ to $Q$ will be denoted by $\text{Mor}(P,Q)$.

The "Stone-Cech" compactification for totally order-disconnected spaces is constructed as follows. Let $\mathbf{2}$ be the space $\{0,1\}$ endowed with the discrete topology and ordered by $0<1$. For any TOD $P$ we set $C^*(P) = \text{Mor}(P,\mathbf{2})$. Note that any product of $\mathbf{2}$ is a TOD when endowed with the componentwise ordering and the product topology. We define the evaluation map $$e_P : P \to \mathbf{2}^{C^*(P)}$$ by $e(p):f\in C^*(P) \mapsto f(p) \in \mathbf{2}$ for all $p\in P$ and set $$\beta_\mathbf{2}(P) := \text{cl}(\text{im}(e_P)),$$ as it is done with the usual Stone-Cech compactification. (It turns out that this construction is indeed a compactification in the category of TODs.)

We say that a TOD $P$ is *well-behaved* if the following is true: if $U$ is a clopen up-set of $P$ then $\downarrow U := \{p \in P: p\leq u \text{ for some } u\in U\}$ is clopen.

**Question**: If $P$ is well-behaved, is the same true for $\beta_\mathbf{2}(P)$?