Localic maps constructed from open and closed parts

Assembling a localic map f\colon L\to M from localic maps f_i\colon S_i\to M, i\in J, defined on closed resp. open sublocales (J finite in the closed case) follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of  preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead,  we present  simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper.

Keywords:

frame , locale , sublocale , sublocale lattice , open sublocale , closed sublocale , localic map , preimage , Boolean frame , linear frame