Separation axioms and frame representation of some topological facts
Similarly as the sobriety is essential for representing continuous maps as frame homo-morphisms, also other separation axioms play a basic role in expressing topological phenomena in frame language. In particular,TD is equivalent with the correctness of viewing subspaces as sublocates, or with representability of open or closed maps as open or closed homomorphisms. A weaker separation axiom is equivalent with an algebraic recognizability whether the intersection of a system of open sets remains open or not. The role of sobriety is also being analyzed in some detail.