Research on the Finite Open Coverage Theorem and Its Application

有限开覆盖定理通过化“无限和运算”为“有限和运算”的方式，将一个局部成立的性质推广到在一个闭区间上整体成立。本文从整体与局部的关系分析了有限开覆盖定理的内涵，并给出具体例子分析此定理在实现整体性质与局部性质相互转化方面以及在实际问题中的应用。The finite open cover theorem extends a property that holds locally to the entire closed interval by transforming “infinite sum operations” into “finite sum operations.” This paper analyzes the essence of the finite open cover theorem from the perspective of the relationship between the global and the local. It provides concrete examples to illustrate how the theorem facilitates the mutual transformation of global and local properties, as well as its applications in practical problems.