中文摘要：

为了解决有限集上的拓扑计数问题，对拓扑的结构性质进行了分析研究；针对穷举法中庞大的计算量，提出了一种集簇式递推算法，以n-1元集合的拓扑为基础，以扩展子集为线索来构建n元集合的拓扑；与传统的穷举法相比，大大降低了时间复杂度，减少了计算时间，并通过实验验证了算法的有效性．

英文摘要：

This study focuses on the solution to the topologies counting problem on a finite set from the perspective of topological structure properties. Based on the topologies on a set of 11-1 elements, a recursive algorithm which constructs the topologies on a set of n elements is put forward following the extended subset. Compared with the traditional method, the algorithm expedites and simplifies the computing. Most importantly, its effectiveness has been verified through experiments.