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arXiv · 2111.14380

The language of pre-topology in knowledge spaces

Fucai Lin·Xiyan Cao·Jinjin Li
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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.

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Fucai Lin, Xiyan Cao, Jinjin Li. 2021-11-29. The language of pre-topology in knowledge spaces. https://arxiv.org/abs/2111.14380

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