Topology via Logic

  title={Topology via Logic},
  author={Steven J. Vickers},
1. Introduction 2. Affirmative and refutative assertions 3. Frames 4. Frames as algebras 5. Topology: the definitions 6. New topologies for old 7. Point logic 8. Compactness 9. Spectral algebraic locales 10. Domain theory 11. Power domains 12. Spectra of rings Bibliography. 
Towards a Duality Result in Coalgebraic Modal Logic
  • B. Jacobs
  • Computer Science, Mathematics
  • CMCS
  • 2000
This paper introduces (back-and-forth) transformations between coalgebras of simple polynomial functors and certain Boolean algebrAs with operators (BAOs) that take the form of an adjunction. Expand
L O ] 6 J ul 2 01 9 Coalgebraic Geometric Logic
Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for anExpand
Functorial semantics of topological theories
Following the categorical approach to universal algebra through algebraic theories, proposed by F.~W.~Lawvere in his PhD thesis, this paper aims at introducing a similar setting for general topology.Expand
Modal Logic and the Vietoris Functor
In [16], Esakia uses the Vietoris topology to give a coalgebra-flavored definition of topological Kripke frames, thus relating the Vietoris topology, modal logic and coalgebra. In this chapter, weExpand
Categorically algebraic topology versus universal topology
  • S. Solovyov
  • Mathematics, Computer Science
  • Fuzzy Sets Syst.
  • 2013
It is shown that a concrete category is fibre-small and topological if and only if it is concretely isomorphic to a subcategory of a category of catalg topological structures, which is definable by topological co-axioms. Expand
Clausal Logic and Logic Programming in Algebraic Domains
A natural declarative semantics and a fixed-point semantics for disjunctive logic programs, and their equivalence are introduced, and it is applied to give both a syntax and semantics for default logic in any coherent algebraic dcpo. Expand
Motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of topological system introduced by S. Vickers, providing anExpand
A pr 2 01 9 Heyting Algebra and G ö del Algebra vs . various Topological Systems and Esakia Space : a Category Theoretic Study
In this paper intuitionistic topological system and its properties have been introduced. Categorical interrelationships among Heyting algebra, Gödel algebra, Esakia space and proposed intuitionisticExpand
Topology, domain theory and theoretical computer science
Abstract In this paper, we survey the use of order-theoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the applications ofExpand
Operational domain theory and topology of a sequential programming language
  • M. Escardó, W. Ho
  • Computer Science, Mathematics
  • 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05)
  • 2005
This work works with an operational notion of compact set and shows that total programs with values on certain types are uniformly continuous on compact sets of total elements and applies this to prove the correctness of non-trivial programs that manipulate infinite data. Expand