Definition:Regular Locale

Definition

Let $L = \struct {S, \preceq}$ be a locale.

Let $\eqslantless$ denote the well inside relation on $L$.

Then $L$ is said to be a regular locale if and only if:

$\forall a \in S : a = \sup \set {b \in S : b \eqslantless a}$

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {III}$: Compact Hausdorff Spaces, $\S1.1$