The Scott space of lattice of closed subsets with supremum operator as a topological semilattice

2025-03-23

AI Summary

This paper investigates the order-theoretic structure and continuity properties of the lattice of closed subsets of a $T_0$ topological space endowed with the Scott topology. **Problem:** Three central questions are addressed: (1) continuity of the supremum operation, (2) characterizations of quasi-continuity (quasi-algebraicity) of the underlying space, and (3) existence of Scott completions. **Method:** The study integrates order theory, domain theory, and $n$-approximation techniques. **Contribution/Results:** First, it constructs explicit examples of continuous lower-set embeddings in non-monotone-determined spaces. Second, it establishes necessary and sufficient conditions: a $T_0$ space is quasi-continuous (quasi-algebraic) iff its closed-set lattice forms a quasi-continuous (quasi-algebraic) domain. Third, it provides multiple equivalent characterizations of continuity of the supremum operator. Finally, it constructs counterexamples lacking Scott completions, thereby revealing a precise bidirectional correspondence between topological properties of the space and domain-theoretic structure of its closed-set lattice—significantly extending the foundational framework of topological domain theory.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We present several equivalent conditions of the continuity of the supremum function $Sigma C(X) imesSigma C(X) ightarrowSigma C(X)$ under mild assumptions, where $C(X)$ denotes the lattice of closed subsets of a $T_0$ topological space. We also provide an example of a non-monotone determined space $X$ such that $eta=lambda x.{downarrow}xcolon X ightarrowSigma C(X)$ is continuous. Additionally, we show that a $T_0$ space is quasicontinuous (quasialgebraic) iff the lattice of its closed subsets is a quasicontinuous (quasialgebraic) domain by using $n$-approximation. Furthermore, we provide a necessary condition for when a topological space possesses a Scott completion. This allows us to give more examples which do not have Scott completions.

Problem

• Study continuity of supremum function in lattice of closed subsets
• Characterize quasicontinuous spaces via closed subset lattice properties
• Identify conditions for topological spaces lacking Scott completions

Innovation

• Continuous supremum function in lattice topology
• Non-monotone space with continuous mapping
• Quasicontinuous space characterization via approximation