A Lattice-Theoretic approach to arbitrary real functions on frames

Abstract

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 on L. This approach mimicks the situation one has with a T₁-space X, where the lattice F(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T₁-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T₀- spaces.

Mathematics Subject Classification (2010): Primary: 06D22; Secondary: 26A15, 54C30, 54D15.

Key words: Frame, locale, frame of reals, continuous real function, order complete, Dedekind-MacNeille completion, semicontinuous real function, partial real function, Haus-dorff continuous real function.