# Enriched closure spaces as a novel framework for domain theory

## TL;DR

This paper introduces enriched closure spaces as a new framework extending continuous lattices and domains, providing a generalized approach to domain theory with novel characterizations and conditions.

## Contribution

It develops the concept of enriched closure spaces, defines a way-below relation, and establishes conditions for continuity and the interpolation property in this new framework.

## Key findings

- Characterization of continuity in enriched closure spaces
- Necessary and sufficient conditions for the interpolation property
- Link between continuity and topology of open subsets

## Abstract

We propose a generalization of continuous lattices and domains through the concept of enriched closure space, defined as a closure space equipped with a preclosure operator satisfying some compatibility conditions. In this framework we are able to define a notion of way-below relation; an appropriate definition of continuity then naturally follows. Characterizations of continuity of the enriched closure space and necessary and sufficient conditions for the interpolation property are proved. We also draw a link between continuity and the possibility for the subsets that are open with respect to the preclosure operator to form a topology.