A Boolean extension of a frame and a representation of discontinuity

Published in: Quaestiones Mathematicae
Volume 40, issue 8, 2017, pages: 1111–1125
DOI: 10.2989/16073606.2017.1348399
Author(s): Jorge PicadoCMUC, Department of Mathematics, Portugal, Aleš PultrDepartment of Applied Mathematics and ITI, MFF, Czech Republic


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.

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