# Topology via Logic

@inproceedings{Vickers1989TopologyVL, title={Topology via Logic}, author={Steven J. Vickers}, year={1989} }

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.

#### 454 Citations

Towards a Duality Result in Coalgebraic Modal Logic

- 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

- 2019

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 an… Expand

Functorial semantics of topological theories

- Mathematics
- 2015

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

- Mathematics
- 2014

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, we… Expand

Categorically algebraic topology versus universal topology

- 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

- Computer Science
- Inf. Comput.
- 2001

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

ON ALGEBRAIC AND COALGEBRAIC CATEGORIES OF VARIETY-BASED TOPOLOGICAL SYSTEMS

- Mathematics
- 2011

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 an… Expand

A pr 2 01 9 Heyting Algebra and G ö del Algebra vs . various Topological Systems and Esakia Space : a Category Theoretic Study

- 2019

In this paper intuitionistic topological system and its properties have been introduced. Categorical interrelationships among Heyting algebra, Gödel algebra, Esakia space and proposed intuitionistic… Expand

Topology, domain theory and theoretical computer science

- Mathematics
- 1998

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 of… Expand

Operational domain theory and topology of a sequential programming language

- 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