Then Supp A is the topology of the Euclidean real line. The previous example doesn't fit in the above characterization as it is not compact. If we add an artificial bottom interval, then we get a compact stably locally compact locale. It is going to appear soon. Perfect double negation on the topology of a space removes the partial points of the space in many cases.
Any clue is very much appreciated.
In another direction, double negation takes us from intuitionistic logic to classical logic.