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.

翻译：我们系统地研究学前空间理论的一些基本特性,如基础前、亚空间前、分离的轴心、连接等。在知识结构理论中,先天学也被称为知识空间。我们讨论了知识空间理论中前天学分离的轴心语言、亚历山德罗霍夫空间和准正态空间的关系,以及将前天学空间密度应用于知识空间初级项目。我们特别给出了技能多图的特征描述,即描述知识结构是一个知识空间,当每个项目具有有限的多种能力时,可以回答“cite{falmagne2011年学习}”或“cite{XGLJ}中的问题;此外,我们给出一种算法,为任何有限的知识空间寻找一组原子初级项目。