Lindelöf tightness and the Dedekind-MacNeille completion of a regular σ-frame.

Tightness is a notion that arose in an attempt to understand the reverse reflection problem: given a monoreflection of a category onto a subcategory, determine which subobjects of an object in the subcategory reflect to it — those which do are termed tight. Thus tightness can be seen as a strong density property. We present an analysis ofλ-tightness, tightness with respect to the localic Lindel¨of reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regularσ-frameAis a frame. Moreover, the embedding ofAin its normal completion is the Bruns-Lakser injective hull ofAin the category of meet semilattices and semilattice homomorphisms. Since every regularσ-frame is the cozero part of a regular Lindel¨of frame, this result points towardsλ-tightness. For any regular Lindel¨of frameL, the normal completion of CozLembeds inLas the sublocale generated by CozL. Although this completion is clearly contained in every sublocale having the same cozero part asL, we show by example that its cozero part need not be the same as the cozero part asL. We prove that a sublocaleSisλ-tight inLiffShas the same cozero part asL. The aforementioned counterexample shows that the completion of CozLis not alwaysλ-tight inL; on the other hand, we present a large class of locales for which this is the case. [ABSTRACT FROM PUBLISHER]