Dear toposophers, Every sober space has a smallest dense perfect subspace. In many cases this is precisely the subspace of maximal points in the specialization order. In general it is larger than that. This is approached via locales. Every locale A has a smallest dense perfect sublocale, which is spatial if A is. It is obtained by the perfect coreflection of the double negation nucleus. The construction seems to generalize from locales to toposes. But what does it produce (instead of subspaces of maximal points)? And what is its logical content? Could toposes of sheaves help one to understand this? Thanks in advance for any clue. Details follow. Martin Escardo --------------------------------------------------------------------- http://www.dcs.ed.ac.uk/home/mhe --------------------------------------------------------------------- Every finite T_0 space has a smallest dense subspace, namely its subspace of maximal points in the specialization order. And every locale has a smallest dense sublocale, induced by the double negation nucleus, as it is well known. But it is also well known that this is hardly ever spatial, even if the given locale is. I have considered a modification of this construction. Let's say that a nucleus on a locale is perfect if it preserves directed joins. If we say that a perfect map is a continuous map f:A->B such that the right adjoint f_*:O(A)->O(B) of the frame map f^*:O(B)->O(A) preserves directed joins, then a nucleus is perfect iff it is induced by a perfect map. Let's also say that a sublocale is perfect if the inclusion is perfect. It turns out that every locale has a smallest dense perfect sublocale. Classically, one can show that spatial locales are closed under perfect sublocales. Hence every sober space has a smallest dense perfect sober subspace. This goes as follows. One knows that the set NA of nuclei on a locale A, with the pointwise ordering, is a frame. Denote by FA the set of perfect nuclei. Then one can show that FA is a subframe of NA. (The join of a set of perfect nuclei is computed by taking the pointwise (directed) join of the finite compositions of the given nuclei.) [[Digression: F is functorial on the category of compact and stably locally compact locales with perfect maps. For such a locale A, the locale FA is compact regular, and if A is compact regular then FA=A. For example, if A is the topology of lower semicontinuity (=Scott topology) on the unit interval, then FA is the Euclidean topology on the unit interval. But this is another story.]] Thus, there is a coreflective adjunction between FA and NA, which in one way includes FA into NA and in the other assigns to each nucleus in NA the join of the perfect nuclei below it. (If A is compact and stably locally compact then there is a simple formula for the perfect coreflection of nuclei on A.) Thus the smallest dense perfect sublocale is induced by the perfect coreflection of the double negation nucleus. But what is it? Let's call it the support of A and denote it by Supp A. The support always contains the subspace of maximal points (oh, I should say Max Pt A is included in Pt Supp A---but I'll refer to a locale as space for terminological simplicity). In many cases it consists exactly of the maximal points. For instance, if A is compact and stably locally compact then this is the case iff the subspace of maximal points is compact, and in this case Supp A is a compact regular locale. (I observe at this point that compact stably locally compact locales are closed under perfect sublocales). Some examples and counterexamples can be useful: (i) Let A be the Scott topology induced by the prefix order on finite and infinite sequences over {0,1}. Then Supp A is the topology of Cantor space. (ii) Same but sequences over natural numbers. Then Supp A = A, and not Baire space as one could expect from the previous example, because Baire space fails badly to be locally compact. (iii) Let A be the Scott topology on compact real intervals ordered by reverse inclusion. Then Supp A is the topology of the Euclidean real line. (iv) The previous example doesn't fit in the above characterization as it is not compact. If we add an artificial bottom interval, then we get a compact stably locally compact locale. But Supp A is then the topology of the Euclidean real line with a bottom element in the specialization order---bottom doesn't get removed in this case. (v) Let UA be the upper power locale of a compact regular locale A. Then Supp UA = A. This again fits in the above characterization. (The above claims [[except the ones in double brackets]] are proved in the paper "Properly injective spaces and functions spaces", which should have been called "Perfectly injective spaces and function spaces". It is going to appear soon. Meanwhile it is available at http://www.dcs.ed.ac.uk/home/mhe/pub/papers/injective.ps.gz) -------------------------- Some questions to finish: -------------------------- Perfect double negation on the topology of a space removes the partial points of the space in many cases. What does it do to toposes? Any clue is very much appreciated. In another direction, double negation takes us from intuitionistic logic to classical logic. Is that behind any correlation intuitionistic logic<->partiality, classical logic<->totality? Can we make this precise by taking toposes of sheaves? The inclusion s:A->Supp A, being a continuous map, induces a geometric morphism S=Shv(s):Shv(A)->Shv(Supp A). Since s_* preserves directed joins, one could guess that S_* preserves directed colimits. Would that be the case? Have these "perfect" geometric morphisms been studied?