SAMS Subject Classification Number: 4, 13, 34

One of the monads described in [2] by H. Simmons is the prime open filter monad whose algebras were shown to be the class of stably compact spaces together with proper maps. Following Simmons and starting with the fact that the coalgebras of the ideal functor are precisely the stably compact frames and proper frame maps [1], we give an alternative description of the relationship between the prime open filter monad and the ideal functor. We shall give a new proof of the equivalence between stably compact spaces and stably compact frames and how this equivalence reduces to that of compact Hausdorff spaces and compact regular frames.

[1] B. Banaschewski and G.C.L. Brümmer, Stably continuous frames, Math. Proc. Camb. Phil. Soc. 104 No 7 (1988), 7–19.

[2] H. Simmons, A couple of triples, Top. Appl. 13 (1982), 201–203.