Modular matrix calculations in the finite topological spaces

Abstract

When performing important calculations in a finite topological space (FTS), matrix calculation methods are more accurate and convenient than traditional methods. However, even when dealing with relatively small subsets involved in the calculations, all elements of the entire space are necessary. This leads to significant time and space waste in practical applications. Therefore, we introduce a modular calculation method as a crucial improvement. Our motivation is as follows: the topological space being processed is divided into modules, ensuring that when any subset is involved in the calculations, only relevant modules are considered instead of the entire space, while ensuring the same result. In addition, the subsets are further divided into smaller subsets within the relevant modules for calculation, greatly reducing the calculation scope and improving the computational efficiency and accuracy. Based on the modularization of the topological space, we propose a modular matrix calculation method and conduct a detailed study of it. Finally, we provide some examples to demonstrate the modular calculation method and modular matrix calculation method.