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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 18 (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 11 (8 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.
Tholen: A functional approach to general topology
 Categorical Foundations (Cambridge
, 2004
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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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Cited by 1 (1 self)
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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.
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.)
GDP Festschrift ENTCS, to appear Abstract A Convenient Category of Domains Dedicated to Gordon Plotkin on the occasion of his 60th birthday
"... 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, provides a mod ..."
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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, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains with such properties. 1
NONCOMMUTATIVE TDUALITY THE DYNAMICAL DUALITY THEORY AND 2DIMENSIONAL EXAMPLES
"... Abstract. A duality theory of bundles of C∗algebras whose fibres are twisted transformation group algebras is established. Classical Tduality is obtained as a special case, where all fibres are commutative tori, i.e. untwisted group algebras for Zn. Our theory also includes the bundles considered ..."
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Abstract. A duality theory of bundles of C∗algebras whose fibres are twisted transformation group algebras is established. Classical Tduality is obtained as a special case, where all fibres are commutative tori, i.e. untwisted group algebras for Zn. Our theory also includes the bundles considered by Mathai and Rosenberg in their work on noncommutative Tduals, in which they allow twisted group algebras on one side of the duality.