# Sharp Elements and Apartness in Domains

@inproceedings{Jong2021SharpEA,
title={Sharp Elements and Apartness in Domains},
author={Tom de Jong},
booktitle={MFPS},
year={2021}
}
• T. D. Jong
• Published in MFPS 28 December 2021
• Mathematics
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges–V…

## References

SHOWING 1-10 OF 43 REFERENCES

### Predicative Aspects of Order Theory in Univalent Foundations

• Mathematics
FSCD
• 2021
The first main result is that nontrivial (directed or bounded) complete posets are necessarily large and weak propositional resizing holds, and it is shown that nontribial locally small δV -complete posets necessarily lack decidable equality.

### Domain Theory in Constructive and Predicative Univalent Foundations

• Computer Science
CSL
• 2021
This work further considers algebraic and continuous dcpos, and construct Scott's $D_\infty$ model of the untyped $\lambda$-calculus, which gives a large, locally small, algebraic dcpo with small directed suprema.

### Exhaustible Sets in Higher-type Computation

A complete description of exhaustible total sets is obtained by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces and shows that they are precisely the computable images of the Cantor space.

### Domain theory

• Mathematics
LICS 1995
• 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

### Predicative theories of continuous lattices

The paper introduces strong proximity join-semilattice, a predicative notion of continuous lattice which arises as the Karoubi envelop of the category of algebraic lattices and shows that this structure naturally corresponds to the notion ofContinuous lattice in the predicative pointfree topology.

### Inductively generated formal topologies

• Philosophy, Mathematics
Ann. Pure Appl. Log.
• 2003

### Constructive Domains with Classical Witnesses

• Mathematics, Computer Science
Log. Methods Comput. Sci.
• 2021
A constructive theory of continuous domains from the perspective of program extraction is developed, and a representation theorem is proved that precisely delineates the class of representable continuous functions.

### A type-theoretic interpretation of constructive domain theory

It is shown that a theory of domains can be developed within a well-defined fragment of (total) type theory, an important step toward constructing a model of all of partial type theory (type theory extended with general recursion) inside total type theory.