Results 1  10
of
11
Properly injective spaces and function spaces
 TO APPEAR IN TOPOLOGY AND ITS APPLICATIONS
, 1997
"... Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous ..."
Abstract

Cited by 28 (12 self)
 Add to MetaCart
Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial onepoint space or j is a proper map in the sense of Hofmann and Lawson. In order to avoid the ambiguous expression “proper subspace embedding”, we refer to proper maps as finitary maps. We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces. These coincide with the retracts of upper power spaces of sober spaces. In the full subcategory of locally compact sober spaces, these are known to be the continuous meetsemilattices. In the full subcategory of stably locally compact spaces these are again the continuous lattices. The above characterization of the injective spaces over finitary embeddings is an instance of a general result on injective objects in posetenriched categories with the structure of a KZmonad established in this paper, which we also apply to various full subcategories closed under the upper power space construction and to the upper and lower power locale monads. The above results also hold for the injective spaces over dense subspace embeddings (continuous Scott domains). Moreover, we show that every sober space has a smallest finitary dense sober subspace (its support). The support always contains the subspace of maximal points, and in the stably locally compact case (which includes densely injective spaces) it is the subspace of maximal points iff that subspace is compact.
Locales Are Not Pointless
 Theory and Formal Methods 1994: Proceedings of the Second Imperial College Department of Computing Workshop on Theory and Formal Methods, Mller
, 1994
"... The KripkeJoyal semantics is used to interpret the fragment of intuitionistic logic containing ; ! and 8 in the category of locales. An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be pr ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
The KripkeJoyal semantics is used to interpret the fragment of intuitionistic logic containing ; ! and 8 in the category of locales. An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be proved as single formal theorems. Openness and properness of maps between locales are characterized by descriptions using the logic, and it is proved that a locale is open iff its lower powerlocale has a greatest point. The entire account is constructive and holds for locales over any topos. 1
Functionspace compactifications of function spaces
 Topology Appl
"... If X and Y are Hausdorff spaces with X locally compact, then the compactopen topology on the set C(X, Y) of continuous maps from X to Y is known to produce the right functionspace topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for an ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
If X and Y are Hausdorff spaces with X locally compact, then the compactopen topology on the set C(X, Y) of continuous maps from X to Y is known to produce the right functionspace topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X, Z) has a compact Hausdorff topology whose relative topology on C(X, Y) is the compactopen topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X, Y) under the compactopen topology is embedded into the Vietoris hyperspace V(X × Y). (2) The space of realvalued continuous functions on a locally compact Hausdorff space under the compactopen topology is embedded into a compact Hausdorff space whose points are pairs of extended realvalued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.
PointSensitive and PointFree Patch Constructions
"... Using the category of frames we consider various generalizations of the patch space of a topological spaces. Some of these are old and some are new constructions. We consider how these variants interact and under what circumstances they agree. Contents 1 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Using the category of frames we consider various generalizations of the patch space of a topological spaces. Some of these are old and some are new constructions. We consider how these variants interact and under what circumstances they agree. Contents 1
Spatiality of the patch frame
"... The Scott continuous nuclei form a subframe of the frame of all nuclei. This subframe is referred to as the patch frame. In this short note we show that the patch of a topology is isomorphic to a finer topology on the same set. We also give a pointset construction of the patch topology of a sober s ..."
Abstract
 Add to MetaCart
The Scott continuous nuclei form a subframe of the frame of all nuclei. This subframe is referred to as the patch frame. In this short note we show that the patch of a topology is isomorphic to a finer topology on the same set. We also give a pointset construction of the patch topology of a sober space.
Compactly generated Hausdorff locales
, 2005
"... We say that a Hausdorff locale is compactly generated if it is the colimit of the diagram of its compact sublocales connected by inclusions. We show that this is the case if and only if the natural map of its frame of opens into the second Lawson dual is an isomorphism. More generally, for any Hausd ..."
Abstract
 Add to MetaCart
We say that a Hausdorff locale is compactly generated if it is the colimit of the diagram of its compact sublocales connected by inclusions. We show that this is the case if and only if the natural map of its frame of opens into the second Lawson dual is an isomorphism. More generally, for any Hausdorff locale, the second dual of the frame of opens gives the frame of opens of the colimit. In order to arrive at this conclusion, we generalize the Hofmann–Mislove–Johnstone theorem and some results regarding the patch construction for stably locally compact locales. Key words. Compactly generated Hausdorff locale, kspace, Lawson duality, Hofmann–Mislove–Johnstone theorem, patch construction.
2.3 The Point Space............................ 17
"... 1 A dummy first page If you want to print 2up, run of from page 2 (the next page). This will get the book page number in the correct corner. The use of the ..."
Abstract
 Add to MetaCart
1 A dummy first page If you want to print 2up, run of from page 2 (the next page). This will get the book page number in the correct corner. The use of the
Supervisor
, 1996
"... PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation ..."
Abstract
 Add to MetaCart
PCF extended with real numbers: a domaintheoretic approach to higherorder exact real number computation