On a topological invariant of finite topological spaces and enumerations
On a topological invariant of finite topological spaces and enumerations
复制标题
关于有限拓扑空间和枚举的拓扑不变量
DOI：
10.21099/tkbjm/1496161830
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发表时间：
1992
影响因子：
0.7
通讯作者：
S. Ochiai
中科院分区：
文献类型：
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作者：
S. Ochiai
设T（n）是n元集合上拓扑的个数。如何确定这个数是一个古老而又困难的问题。许多作者攻击这个问题，到目前为止，确定了这个数字T（n）的小值n。最近，M。Erne宣称他通过使用一个新的约化公式[4]确定了对所有n^Yi的这个数T（n）。本文定义了一个与T（n）的确定有关的拓扑不变量，研究了它的性质，并进行了一些与它有关的计算，而不是为了确定T（n）.如果X是一个n元的有限拓扑空间，则X由包含它的每个点x的极小开集Ux确定。如果UxQUy成立，那么我们在X上定义关系< by x<y。这个关系是自反的和传递的，所以（X，<）是一个拟序集。相反，对于给定的拟序集（X，<），如果我们定义Ux为Ux={y\y<x}，我们可以将有限拓扑空间与极小基{Ux \x<bX}联系起来。这给出了X上的所有拓扑与X上的所有拟序之间的一一对应，并且还导出了X上的所有To拓扑与X上的所有偏序之间的一一对应[1，p，28]，[2，p. 14]，[7，p. 142]。X上的所有拓扑与所有n元标号可迁有向图之间存在一一对应关系[5]。进一步地，在n元集合上的所有To拓扑，所有秩为n的有限分配格L，所有n元标号可迁无圈有向图之间存在一一对应关系[5]，[11].设T0（n）表示n元集合上所有To拓扑的个数。J.W. Evance，F. Harary和MS林恩证明了
Let T(n) be the number of topologies on a set with n-element. It is an old and difficultproblem to determine this number. Many authors attacked this problem and so far determined this number T(n) for small value n. Recently, M. Erne pronounced that he determined thisnumber T(n) for all n^Yi, by using a new reduction formula [4]. In this paper, we will define a topological invariant which relates to the determination of T{n), investigate its property and make some computations in connection with its invariant rather than intend to determine T{n). If X is a finitetopological space with n-element, then X is determined by the minimal open set Ux containing each of its point x. If UxQUy holds, then we define the relation < by x<y on X. This relation ^ is reflexive and transitive and so (X, <) is a quasi ordered set. Conversely, with a given quasi ordered set (X, <), if we define Ux by Ux={y\y<x}, we can associate the finitetopological space with the minimal base {Ux \x<bX}. This gives a one to one correspondence between all topologies on X and all quasi orders on X, and also induces a one to one correspondence between all To topologies on X and allpartial orders on X [1, p, 28], [2, p. 14], [7, p. 142]. It is also well known that there is a one to one correspondence between all topologies on X and all the labeled transitive digraphs with nelement [5]. Furthermore there is a one to one correspondence between all To topologies on a set with n-element, all finitedistributive lattices L of rank n, all the labeled transitive acyclic digraphs with n-element [5], [11]. Let T0(n) denote the number of all To topologies on a set with n-element. J.W. Evance, F. Harary and M.S. Lynn proved that