Localic real functions: A general setting

Abstract

In pointfree topology the lattice-ordered ring of all continuous real functions on a frame L has not been a part of the lattice of all lower (or upper) semicontinuous real functions on L just because all those continuities involve different domains. This paper demonstrates a framework in which all those continuous and semicontinuous functions arise (up to isomorphism) as members of the lattice-ordered ring of all frame homomorphisms from the frame L ( R ) of reals into S ( L ) , the dual of the co-frame of all sublocales of L . The lattice-ordered ring Frm ( L ( R ) , S ( L ) ) is a pointfree counterpart of the ring R X with X a topological space. We thus have a pointfree analogue of the concept of an arbitrary not necessarily ( semi) continuous real function on L . One feature of this remarkable conception is that one eventually has: lower semicontinuous + upper semicontinuous = continuous. We document its importance by showing how nicely can the insertion, extension and regularization theorems, proved earlier by these authors, be recast in the new setting.