Open locale

An open, or overt, locale has a positivity map $\mathrm{Pos}:\mathcal{O}(X)\to \Omega$ left adjoint to the scalar inverse image ${!}^{\ast }:\Omega \to \mathcal{O}(X)$, satisfying Frobenius reciprocity. Thus $a\le {!}^{\ast }\mathrm{Pos}(a)$ and positivity preserves arbitrary joins. Classically every locale is overt, with positivity meaning $a\ne 0$.

A positive open locale is totally connected when every positive open of a locale is connected. Under classical logic its positive opens form a completely prime filter, hence specify a dense point. Positivity of the whole locale is essential: the empty locale satisfies the condition on positive opens vacuously but has no point.

An open is positive when its positivity truth value holds. Positivity gives a constructive witness of inhabitedness and must not be replaced by double-negated nonemptiness.