N-Compact Frames and Applications

Abstract

<p>This thesis is a study of what it means for a frame to be N-compact. We find that the frame analogues of equivalent conditions defining N-compact spaces are no longer equivalent in the frame context; one must be careful in deciding what the appropriate frame notion is. We show that it is the assumption of a choice principle (the Axiom of Countable Choice) which provoke9 this departure from the spatial situation.</p> <p>We analyze the several possibilities and show how it is the 'H-N-compact' frames which best embody the notion of N-compactness. We develop the theory and construct the H-N-compactification, which uses a frame-theoretic version of the classical ultrafilter formulation of the spatial N-compactification. We use this compactification to show how these frames relate to the other 'N-compact' frames. Along the way we construct a 0-dimensional Lindelōf co-reflection, and show how this relates to the H·N-compactification.</p> <p>Recent works in Abelian group theory have employed the groups C(X, Z) in the study of reflexivity and duality. The N-compact spaces are important in this regard because of a theorem of Mrówka which shows how a group homomorphism from C(X, Z) to Z is determined on a small part of X, if X is N-compact. We use the H-N-compact frames to lift thus to a result about any group of global sections of a sheaf of Abelian groups. We then are able to give a sufficient condition for the local reflexivity of a sheaf to lift to global reflexivity; it is enough that the frame is H-N-compact. We show that the groups known to be reflexive (in ZFC) each appear as a group of global sections of some sheaf on an N-compact frame, or as the dual of such a group of sections. We can then use our generalized Mrówka's Theorem to establish their reflexivity.</p> <p>In the final chapter we apply the techniques of Chapter 1 to the study of realcompact frames. These have been studied, but the definition usually given is quite restrictive. We construct the H-realcompactification and develop enough of the basic theory of H-realcompact frames to justify proposing that these be thought of as the realcompact frames.</p>