Continuity of Powerspaces Structures in Directed Spaces

TL;DR

This paper investigates the continuity properties of powerspaces in directed spaces, demonstrating that certain powerspaces of continuous spaces retain continuity, which is important for modeling nondeterministic programming semantics.

Contribution

It establishes that directed lower, upper, and convex powerspaces of continuous spaces are themselves continuous, advancing the understanding of powerspace structures in directed spaces.

Findings

Directed powerspaces of continuous spaces are continuous.

Continuity of powerspaces supports semantic modeling of nondeterministic programs.

Provides theoretical foundation for powerspace constructions in directed topology.

Abstract

Powerspaces of directed spaces play an important role in modeling the semantics of nondeterministic functional programming languages. The notions of upper,lower and convex powerspace of a directed space are defined by the way of free algebras[25]. In this paper, We study the continuity of power structures of directed spaces and show that the directed lower powerspaces, directed upper powerspaces and directed convex powerspaces of continuous spaces are continuous spaces.