Important matrix computations in finite topological spaces

Because of their wide applications in many areas, finite topological spaces have been attracting more and more attention from researchers. Especially in some fields such as data mining and computing, interior operators and closure operators play very important roles. However, we all know that it is not easy to find the interior or the closure of a set by definition generally in a finite topological space, even if the space contains very few elements. In this paper, we introduce the concept of base matrix of the topology, then we give the matrix computation formulas of the interior, the closure, the boundary, as well as the Kuratowski 14 sets of any subset in a n-element topological space where n is any positive integer. Moreover, a Generalized 14 Sets Theorem is introduced and the matrix representations of the generalized 14 sets are also given. We finally summarized the calculation steps and demonstrate the computations through an example. If n is very big, the most important merits of the matrix methods is making the computations simpler, more efficient, more accurate, and can be easily implemented by computer programs. So, these results can greatly improve the application scope and the application operability of finite topological spaces in other fields.