A representation of continuous lattices based on closure spaces

Abstract

In this paper, we establish the link between continuous lattices and closure spaces. By generalizing the notion of algebraic closure space to continuous closure space, we show that continuous lattices can be represented by continuous closure spaces, just as algebraic lattices can be represented by algebraic closure spaces. We also introduce the notion of approximable mappings between continuous closure spaces, which produces the category equivalent to that of continuous lattices with Scott-continuous functions. These results demonstrate the capability of closure spaces in representing continuous domains.