Page 178 - Textos de Matemática Vol. 46

168 J. GUTIE´RREZ GARC´IA, J. PICADO, AND A. PULTR
h: L(R) ! L. This was originally done by Banaschewski ([3] – see [27] for fur- ther references). However, the classical theory of real functions, not necessarily continuous, calls for a point-free counterpart as well. An appropriate defini- tion was presented in [15] and developed in subsequent papers (e.g. [18, 6]). A classical (general) real function on a space (X,⌦(X)) is a continuous real function on the discrete space (X,P(X)). The lattice P(X) of all subsets of X has a natural counterpart in S(L)op where S(L) is the co-frame of all sublo- cales of L. Hence, a (general) real function on L can be represented as a frame homomorphism L(R) ! S(L)op.
The present paper is inspired by [26]. Using extensively the technique of sublocales, we present a survey of some facts on point-free real functions. Most of the results are not new; the originality is essentially in the presentation. Our main goal is to show how zero sets may be considered in the localic setting (as zero sublocales) and then how several important notions and results about real functions may be rewritten and directly proved using this tool.
After some necessary preliminaries we introduce the point-free real func- tions and prove a few facts, in particular some results concerning images and preimages of sublocales are discussed. Then, semicontinuous functions and their relation with the continuous ones are mentioned. In the following section, point- free algebraic operations on L(R) are studied, with special attention paid to the addition, multiplication, maximum and minimum. Next we turn to cozero and zero sublocales. The concept of cozero element is a well-known standard topic and its sublocale counterpart is straightforward, but there are no reasonable zero elements while in the context of sublocales we obtain a sensible notion. This approach allows to formulate the basics of the theory in a way very much parallel to the classical book of Gillman and Jerison [12]. We illustrate this in a miscellany of topics.
Regarding general background, we refer to Picado and Pultr [27] for frames and locales and to Banaschewski [3] and Ball and Walters-Wayland [1] for specific information on continuous functions on frames.
1. Preliminaries I. Free constructions
We will work with point-free real numbers as they are usually described in literature, that is, by generators subject to relations. Since the free generators come from a set that is in fact a meet-semilattice (while its elements are used in the free construction simply as elements of a set) we think that it may be useful for the reader to compare the free frames over sets with free frames over semilattices.