The Dedekind reals in abstract Stone duality
- MathematicsMathematical Structures in Computer Science
The core of the paper constructs the real line using two-sided Dedekind cuts, and shows that the closed interval is compact and overt, where these concepts are defined using quantifiers.
An Introduction to Topology and its Applications : a new approach
- Computer Science
This paper exploits some basic experience with computations, scientific measurements and observations, such as those encountered in the use of a scientific calculator, a computer program or a measuring device, to introduce the topic of topology to a reader with little background knowledge of university mathematics.
Synthetic Topology and Constructive Metric Spaces
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…
Notes for mini-course on computational topology
- Computer Science
This document is meant to be a reference guide for a series of five talks given at the Intensive Research Program in Discrete, Combinatorial and Computational Geometry; as such, each of these talks will focus on various uses of computational topology that overlap most directly with algorithms, computational geometry, and graphics.
On the bitopological nature of Stone Duality
Based on the theory of frames we introduce a Stone duality for bitopological spaces. The central concept is that of a d-frame, which axiomatises the two open set lattices. Exploring the resulting…
On the topological aspects of the theory of represented spaces
This work presents an abstract and very succinct introduction to the theory of represented spaces, drawing heavily on prior work by Escardo, Schroder, and others.
Descriptive Set Theory in the Category of Represented Spaces
- Mathematics2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
This work can reformulate DST in terms of endofunctors on the categories of represented spaces and computable or continuous functions and satisfies the demand for a uniform approach to both classic and effective DST.
Computably Based Locally Compact Spaces
- MathematicsLog. Methods Comput. Sci.
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.
A new introduction to the theory of represented spaces
Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable…
SHOWING 1-10 OF 138 REFERENCES
SOBER SPACES AND CONTINUATIONS
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…
The category-theoretic solution of recursive domain equations
- Mathematics18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
The purpose of the present paper is to set up a categorical framework in which the known techniques for solving equations find a natural place, generalizing from least fixed-points of continuous functions over cpos to initial ones of continuous functors over $\omega $-categories.
- MathematicsLICS 1995
bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the…
Comparing Functional Paradigms for Exact Real-Number Computation
It is shown that the type hierarchies coincide up to second-order types, and it is demonstrated that, in the extensional approach, parallel primitives are necessary for programming total first-order functions, but are not needed for second- order types and below.
SUBSPACES IN ABSTRACT STONE DUALITY
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…
Some Topologies for Computations
- Computer Science
The applications of topological and order structures in Theory of Computation, a key aspect of Foundations of Mathematics and of Theoretical Computer Science, has various origins and it is largely…
Power Domains and Predicate Transformers: A Topological View
The specific tasks are to provide a more adequate framework for power-domain constructions; and to show that the connection between (Dijkstra's) weakest preconditions and the Smyth powerdomain, established by Plotkin for the case of flat domains, actually holds in full generality.