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 sucient 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).

Keywords

• Frame
• spatial frame
• locale
• sublocale
• sublocale lattice
• induced sublocale
• T-D -axiom
• scattered space
• hereditarily irresolvable space