From λ-hollow frames to λ-repletions in W: I. The scaffolding of a completely regular frame

Abstract

We hollow out a given completely regular frame L by removing, for an uncountable regular cardinal number κ, the dense κ-cozero elements of L. This we do by passing to the cointersection of the proper dense open κ-cozero quotients of L. The quotient frame is κ-hollow, i.e., it has no proper dense κ-cozero elements, and this passage is a bireflection in the category of frames with skeletal morphisms. Thus each frame L has a descending sequence of dense quotient frames of increasing hollowness, indexed by the uncountable regular cardinal numbers, culminating in the classical minimal dense quotient of L.Our purpose is to shed light on the structure of a given frame L by examining the collection of quotients and sub-κ-frames which arise in its hollowing sequence, to which we refer collectively as the scaffolding of L.