The language of pre-topology in knowledge spaces

TL;DR

This paper explores the foundational properties of pre-topological spaces, also known as knowledge spaces, and their applications in modeling knowledge structures, including characterizations and algorithms for finite cases.

Contribution

It provides a systematic study of pre-topological properties in knowledge spaces, characterizes skill multimaps, and offers algorithms for finite knowledge space analysis.

Findings

Characterization of skill multimaps for knowledge spaces

Algorithm for primary item set in finite knowledge spaces

Relation between Alexandroff spaces and quasi ordinal spaces

Abstract

We systematically study some basic properties of the theory of pre-topological spaces, such as, pre-base, subspace, axioms of separation, connectedness, etc. Pre-topology is also known as knowledge space in the theory of knowledge structures. We discuss the language of axioms of separation of pre-topology in the theory of knowledge spaces, the relation of Alexandroff spaces and quasi ordinal spaces, and the applications of the density of pre-topological spaces in primary items for knowledge spaces. In particular, we give a characterization of a skill multimap such that the delineate knowledge structure is a knowledge space, which gives an answer to a problem in \cite{falmagne2011learning} or \cite{XGLJ} whenever each item with finitely many competencies; moreover, we give an algorithm to find the set of atom primary items for any finite knowledge spaces.