2021/11/29 by Lin, Fucai, Cao, Xiyan, Li, Jinjin
#54B05 #54B10 #54D05 #54D70 #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #FOS: Mathematics #General Topology (math.GN) #Primary 54A05 #secondary 54A25
paper · doi:10.48550/arxiv.2111.14380
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 \citefalmagne2011learning or \citeXGLJ 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.