# The Dedekind reals in abstract Stone duality

@article{Bauer2009TheDR, title={The Dedekind reals in abstract Stone duality}, author={A. Bauer and Paul Taylor}, journal={Mathematical Structures in Computer Science}, year={2009}, volume={19}, pages={757 - 838} }

Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were…

## 29 Citations

A lambda calculus for real analysis

- MathematicsJ. Log. Anal.
- 2010

This is an introduction to ASD for the general mathematician, with application to elementary real analysis, where this language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line.

Interval Analysis Without Intervals

- Mathematics
- 2006

We argue that Dedekind completeness and the Heine–Borel property should be seen as part of the “algebraic” structure of the real line, along with the usual arithmetic operations and relations.…

Ecient Computation with Dedekind Reals

- Mathematics
- 2008

Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts…

Topological domain theory

- Mathematics
- 2008

This thesis presents Topological Domain Theory as a powerful and flexible framework for denotational semantics, and shows that it supports a wide range of semantic constructions, including the standard domain-theoretic construction, computational effects and polymorphism, all within a single setting.

Efficient Computation with Dedekind Reals

- Mathematics
- 2008

Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts…

Homotopy Type Theory: Univalent Foundations of Mathematics

- MathematicsArXiv
- 2013

This book is intended as a first systematic exposition of the basics of the resulting"Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.

Temporal Type Theory

- PhilosophyProgress in Computer Science and Applied Logic
- 2019

This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by…

Synthetic Topology and Constructive Metric Spaces

- Mathematics
- 2010

The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology…

## References

SHOWING 1-10 OF 171 REFERENCES

A lambda calculus for real analysis

- MathematicsJ. Log. Anal.
- 2010

This is an introduction to ASD for the general mathematician, with application to elementary real analysis, where this language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line.

Computably Based Locally Compact Spaces

- MathematicsLog. Methods Comput. Sci.
- 2006

This paper uses the full subcategory of overt discrete objects of ASD to translate computable bases for classical spaces into objects in the ASD calculus, and shows this subcategory to be equivalent to a notion of computable basis for locally compact sober spaces or locales.

Non-Artin Gluing in Recursion Theory and Lifting in Abstract Stone Duality

- Mathematics
- 2003

Stone duality is a radical reformulation of general topology, in which the topology on a space X is not considered as a set carrying an infinitary lattice structure, but as another space that’s the…

SUBSPACES IN ABSTRACT STONE DUALITY

- Mathematics
- 2002

B yabstract Stone duality we mean that the topology or contravariant powerset functor, seen as a self-adjoint exponential Σ (−) on some category, is monadic. Using Beck's theorem, this means that…

Continuity on the real line and in formal spaces

- MathematicsFrom sets and types to topology and analysis
- 2005

Bishop simply modified the definition of continuous function on the real numbers to mean: uniformly continuous on each finite and closed interval, a very successful step, however, it may also lead to difficulties, when going beyond metric spaces.

SOBER SPACES AND CONTINUATIONS

- Mathematics
- 2002

A topological space is sober if it has exactly the points that are dictated by its open sets. We explain the analogy with the way in which computational values are determined by the observations that…

Interval Analysis Without Intervals

- Mathematics
- 2006

We argue that Dedekind completeness and the Heine–Borel property should be seen as part of the “algebraic” structure of the real line, along with the usual arithmetic operations and relations.…

Ecient Computation with Dedekind Reals

- Mathematics
- 2008

Cauchy’s construction of reals as sequences of rational approximations is the theoretical basis for a number of implementations of exact real numbers, while Dedekind’s construction of reals as cuts…