A lattice-theoretic approach to arbitrary real functions on frames

TL;DR

This paper develops a lattice-theoretic framework for modeling arbitrary real functions on frames, extending classical topological concepts to a pointfree setting and providing new definitions for discontinuity and semicontinuity.

Contribution

It introduces a novel lattice-theoretic approach to represent real functions on frames, generalizing semicontinuity and discontinuity concepts from classical topology.

Findings

Models discontinuous functions via Dedekind-MacNeille completion

Defines lattice-theoretic notions of semicontinuity and discontinuity

Provides conservative definitions for T1 and T_D-spaces

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 $T_1$-space $X$, where the lattice $\overline{\mathrm{F}}(X)$ 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 $T_1$-spaces. We also analyze semicontinuity and introduce definitions which are conservative for $T_D$-spaces.