Sublocale

A sublocale is determined by a nucleus on a frame $j$ on $\mathcal{O}(X)$. Its frame consists of the fixed elements of $j$, with inherited meets and regularized joins. Increasing the nucleus makes the sublocale smaller.

The double-negation nucleus on a frame gives a dense sublocale whose frame is a complete Boolean algebra. Its elements are regular open elements, with Boolean joins $\neg \neg {\bigvee }_i{a}_i$.

A sublocale is flat when its nucleus on a frame preserves finite joins, including $0$. Equivalently its fixed elements are closed under ambient finite joins. In particular every flat sublocale is a dense sublocale.

Every map from a flat sublocale $A_j$ of $A$ to a compact locale that is also a regular locale extends uniquely to $A$. On frames the extension iswhere the terms are first regarded as fixed elements in the ambient frame. Regularity proves agreement after applying $j$; compactness reduces arbitrary covers to finite subcovers on a well-inside open, and flatness identifies those finite joins with ambient joins. These facts prove that $g^{\ast }$ preserves arbitrary joins and finite meets.

A sublocale is dense when its closure is the whole locale. For a map $f$, its image is dense when $f^{\ast }(a)=0$ implies $a=0$. Classically, a dense point belongs to every nonzero open.

The closed nucleus $c(a)$ determines the complement of the open $a$. A sublocale with nucleus $j$ is dense precisely when $j(0)=0$, since its closure is the closed sublocale defined by $j(0)$.