Hewitt’s irresolvability and induced Sublocales in spatial frames

Abstract

Sublocales of frames, even those representing subspaces (induced sublo-cales), are typically not complemented in the lattice of all sublocales. We present a necessary and sufficient condition for an induced sublocale to be so, and prove that all induced sublocales are complemented iff the space in question is hereditarily irresolvable, a property slightly weaker than – and in a broad class of spaces equivalent with – scatteredness (under which condition, by Simmons’ result all sublocales are complemented).