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NOTES ON POINT-FREE REAL FUNCTIONS AND SUBLOCALES
JAVIER GUTIE´RREZ GARC´IA, JORGE PICADO, AND ALESˇ PULTR
Dedicated to Manuela Sobral
Abstract. Usingthetechniqueofsublocaleswepresentasurveyofsome known facts (with a few new ones added) on point-free real functions. The subjects treated are, e.g., images and preimages, semicontinuity, algebraic structure (point-free real arithmetics), zero and cozero parts, z- embeddings, z-open and z-closed maps, disconnectivity, small sublocales and supports.
Introduction
The frame of reals, L(R) (“point-free real numbers”), was originally intro- duced by Joyal in an unpublished manuscript [24] and thoroughly studied by Banaschewski in [3] (see also Johnstone [23]). As one might expect, it was not defined as the lattice ⌦(R) of open sets in the standard real line R but as a pri- marily algebraic entity, the free frame generated by pairs of rational numbers (which one can intuitively view as rational intervals) factorized by natural re- lations (see 2.3 below). Under the Axiom of Choice, L(R) is indeed isomorphic with ⌦(R), but the point is to have the frame of point-free reals as a frame in its own right and to be able to avoid choice whenever possible (it should be noted that one can prove in a choice-free way for instance that L(R) is the completion of the frame of rationals or that it is continuous, that is, locally compact, see [3]).
Once one has the frame of real numbers, one can also represent contin- uous real functions on a general frame L, namely as frame homomorphisms
Received: 30 July 2014 / Accepted: 23 December 2014.
2010 Mathematics Subject Classification. Primary 06D22; Secondary 13J25, 26A15, 54C30,
54D15.
Key words and phrases. Frame, locale, sublocale lattice, localic map, frame of reals, real
function, upper semicontinuous, lower semicontinuous, ring of continuous functions, com- pletely separated sublocales, z-embedding, z-open map, z-closed map, perfectly normal frame, small sublocale, support.
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