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A Convenient Category of Domains
 GDP FESTSCHRIFT ENTCS, TO APPEAR
"... We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also su ..."
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Cited by 12 (3 self)
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We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.
The Convergence Approach to Exponentiable Maps
 352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER
, 2000
"... Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the ..."
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Cited by 9 (7 self)
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Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the WhiteheadMichael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.
III A Functional Approach to General Topology
"... In this chapter we wish to present a categorical approach to fundamental concepts of General Topology, by providing a category X with an additional structure which allows us to display more directly the geometric properties of the objects of X regarded as spaces. Hence, we study topological properti ..."
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Cited by 3 (0 self)
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In this chapter we wish to present a categorical approach to fundamental concepts of General Topology, by providing a category X with an additional structure which allows us to display more directly the geometric properties of the objects of X regarded as spaces. Hence, we study topological properties for them, such as Hausdorff separation, compactness and local compactness, and we describe important topological constructions, such as the compactopen topology for function spaces and the StoneČech compactification. Of course, in a categorical setting, spaces are not investigated “directly ” in terms of their points and neighbourhoods, as in the traditional settheoretic setting; rather, one exploits the fact that the relations of points and parts inside a space become categorically special cases of the relation of the space to other objects in its category. It turns out that many stability properties and constructions are established more economically in the categorical rather than the settheoretic setting, leave alone the much greater level of generality and applicability. The idea of providing a category with some kind of topological structure is certainly
TEN TOPOLOGIES FOR 1x7
, 1963
"... THE study of topologies on X x Y is motivated by some outstanding deficiencies of the cartesian, that is the usual, topology on the product of spaces. (Throughout this paper all spaces will be assumed to be Hausdorff.) ..."
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THE study of topologies on X x Y is motivated by some outstanding deficiencies of the cartesian, that is the usual, topology on the product of spaces. (Throughout this paper all spaces will be assumed to be Hausdorff.)
Foundations for Computable Topology
, 2009
"... Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between al ..."
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Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between algebra and geometry as an abstract monadic adjunction that we turn into a new type theory. To this we add an equation that is satisfied by the Sierpiński space, which plays a key role as the classifier for both open and closed subspaces. In the resulting theory there is a duality between open and closed concepts. This captures many basic properties of compact and closed subspaces, despite the absence of any explicitly infinitary axiom. It offers dual results that link general topology to recursion theory. The extensions and applications of ASD elsewhere that this paper survey include a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0, 1] in ASD is compact.