Esakia space

Remarks

In a Priestley space, ${\downarrow} C$ is always closed when $C$ is closed; the extra condition is that it is also open when $C$ is clopen. For clopen up-sets $U$ and $V$ of an Esakia space, the set $X \setminus {\downarrow}(U \setminus V)$ is a clopen up-set, and it is the Heyting implication $U \to V$.

Example: a finite poset $(P, \le)$ with the discrete topology. Every subset is clopen, so the separation condition holds with $U = {\uparrow} x$, and every down-set is clopen.

Non-example: let $X = \set{a_1, a_2, \ldots} \cup \set{\infty, b}$, where $a_n \to \infty$ as in the one-point compactification of $\N$, and $b$ is an isolated point. Order $X$ by $\infty \le b$ and no other strict relations. This is a Priestley space, but $\set{b}$ is clopen and ${\downarrow}\set{b} = \set{\infty, b}$ is not open, since every neighbourhood of $\infty$ contains some $a_n$.