The pair $(Q, \mathscr{ K})$ is a { \it knowledge space} if $\bigcup\mathscr{ K}=Q$ and $\mathscr{ K}$ is closed under union, where $Q$ is a nonempty set and $\mathscr{ K}$ is a family of subsets of $Q$. A knowledge space $(Q, \mathscr{ K})$ is called { \it complementary} if there exists a non-discrete knowledge space $(Q, \mathscr{ L})$ such that the following (i) and (ii) satisfy: (i) for any $q\in Q$, there are finitely many $K_{ 1}, \cdots, K_{ n}\in \mathscr{ K}$ and $L_{ 1}, \cdots, L_{ m}\in \mathscr{ L}$ such that $$(\bigcap_{ i=1}^{ n}K_{ i})\cap (\bigcap_{ j=1}^{ m}L_{ j})=\{ q\};$$ (ii) $\mathscr{ K}\cap \mathscr{ L}=\{ \emptyset, Q\}$. In this paper, the existence of a complementary knowledge space for each knowledge space is proved, and a method of the construction of complementary finite knowledge spaces is given.
Categories: math.GM