求n元集合X上所有拓扑的个数问题，是拓扑学中基本的问题之一，但至今n元集合X上所有拓扑的个数T(n)都没有人能给出一个公式。 目前，研究这一问题主要有两种思路。一是通过建立拓扑与偏序关系的一一对应，将问题转化为求偏序关系的个数； 另一种方法是找到所有拓扑个数与T0拓扑个数的关系,将问题转化为求T0 拓扑的个数.每一种思路都得到了一些成果，但最终都是未能解决。 本文从最基础的拓扑的定义出发，按照同胚拓扑的意义尝试将$n$元集合上的拓扑分类，给出代表拓扑和极大元素的概念，将问题转化为研究极大元素的性质。 较为清楚的情形本文已经给出递归的公式，复杂的情形只是给出了较粗的估计，虽然没有得到具有突出意义的结果，但给出了解决这一问题的另一种思路。

Counting the number of topologies on an n points set X is a basic problem in topology,however, nobody can give a formula of T(n) which express the number of topologies on a n points set X.At present,there are two ways to study this problem.One way is that by founding a one-to-one correspondence between topologies and preorder relations,then the problem is transformed into counting the number of preorder relations; the other way is that by founding a relation between the number of all topologies and the number of T0 topologies, then the problem is transformed into counting the number of T0 topologies. Unfortunately,the problem isn't solved until now though we can gain some achievements in any way above. This dissertation start from the definition of topology,classify the topologies according to the meaning of homeomorphism topology, trying to show the definitions of representative topology and maximal element are given.Then the problem is transformed into discuss the properties of maximal element.This dissertation gives recursive formulas in relatively clear cases and rough estimate in complex situations. This dissertation gives another way to solve the problem though it didn't gain very important result.