A LATTICE-THEORETIC APPROACH TO ARBITRARY REAL FUNCTIONS ON FRAMES

摘要

In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions oil L. This approach mimicks the situation one has with a T-1-space X, where the lattice (F) over bar (X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions oil X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T-1-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T-0- spaces.