The P-frame reflection of a completely regular frame

摘要

We show that every completely regular frame has a P-frame reflection. The proof is straightforward in the case of a Lindelof frame, but more complicated in the general case. The chief obstacle to a simple proof is the important fact that a quotient of a P-frame need not be a P-frame, and we give an example of this. Our proof of the existence of the P-frame reflection in the general case is iterative, freely adding complements at each stage for the cozero elements of the stage before. The argument hinges on the significant fact that frame colimits preserve Lindelof degree. We also outline the relationship between the P-frame reflection of a space X and the topology of the P-space coreflection of X. Although the former frame is generally much bigger than the latter, it is always the case that the P-space coreflection of X is the space of points of the P-frame reflection of the topology on X.