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 higher-order 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 one-point 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, t...

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The Kripke-Joyal 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

If X and Y are Hausdorff spaces with X locally compact, then the compact-open 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 compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X, Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X × Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.

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

Many categories of semantic domains can be considered from an order-theoretic point of view and from a topological point of view via the Scott topology. The topological point of view is particularly fruitful for considerations of computability in classical spaces such as the Euclidean real line. When one embeds topological spaces into domains, one requires that the Scott continuous maps between the host domains fully capture the continuous maps between the guest topological spaces. This property of the host domains is known as injectivity. For example, the continuous Scott domains are characterized as the injective spaces over dense subspace embeddings (Dana Scott, 1972, 1980). From a third point of view, the continuous Scott domains arise as the algebras of a monad (Wyler, 1985). The topological characterization by injectivity turns out to follow from the algebraic characterization and general category theory (Escard'o 1998). In this paper we systematically consider monads that arise ...

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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, k-space, Lawson duality, Hofmann–Mislove–Johnstone theorem, patch construction.

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 point-set construction of the patch topology of a sober space.