Categorical Representations of Continuous Domains and Continuous L-Domains Based on Closure Spaces

Closure space has proven to be a useful tool to restructure lattices and various order structures.This paper aims to provide a novel approach to characterizing some important kinds of continuous domains by means of closure spaces. By introducing an additional map into a given closure space, the notion of F-augmented generalized closure space is presented. It is shown that F-augmented generalized closure spaces generate exactly continuous domains. Moreover, the notion of approximable mapping is identified to represent Scott-continuous functions between continuous domains. These results produce a category equivalent to that of continuous domains with Scottcontinuous functions. At the same time, two subclasses of F-augmented generalized closure spaces are considered which are representations of continuous L-domains and continuous bounded complete domains, respectively.

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