A Boolean extension of a frame and a representation of discontinuity

Abstract

Point-free modeling of mappings that are not necessarily continuous has been so far based on the extension of a frame to its frame of sublocales, mimicking the replacement of a topological space by its discretization. This otherwise successful procedure has, however, certain disadvantages making it not quite parallel with the classical theory (see Introduction). We mend it in this paper using a certain extension Sc(L) of a frame L, which is, a.o., Boolean and idempotent. Doing this we do not lose the merits of the previous approach. In particular we show that it yields the desired results in the treatment of semicontinuity. Also, there is no obstacle to using it as a basis of a point-free theory of rings of real functions; the \ring of all real functions" F(L) = C(Sc(L)) is now order complete.

Mathematics Subject Classication (2010): 06D22, 54C30.

Key words: Frame, locale, subt, regular, sublocale, sublocale lattice, open sublocale,
closed sublocale, real function, lower and upper semicontinuities, lower and upper regularizations.