Does somebody know whether for a complete Heyting algebra (cHa) A such that the specialization order on points is discrete it automatically holds whether for ALL cHa's B the frame morphisms from A to B are ordered discretely by the spezialization order. I know that there are cases (different from complete Boolean algebras) where it holds but does it hold in general ?
Thomas Streicher
It's not true. I expect there are already lots of counterexamples amongst non-trivial pointless locales, but here's something more straightforward. Let D be a non-trivial locale with no global points (global point = locale map from 1 to D) and let D' be its lift (or localification), which has a new bottom point adjoined - by adjoining a new top to its frame. D' has only one global point (the new bottom) and so the specialization order on global points is discrete. However, we have two generalized points of D' at stage D' itself (locale maps from D' to D'), namely the generic point id: D' -> D' and the bottom _|_: D' -> 1 -> D', and they are distinct with _|_ <= id. Steve Vickers.