Taking the covering dimension dim as notion for the dimension of a topological space, we first specify thenumber zdim_{T_0}(n) of zero-dimensional T_0-spaces on {1,...,n}$ and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,\ldots,n} by means oftwo mappings po and P that yieldthe number po(n) of partial orders on {1,...,n} and the set P(n) of partitions of {1,...,n}, respectively. Algorithms for both mappings exist. Assuming one for po to be at hand, we use our specification of zdim_{T_0}(n) and modify one for P in such a way that it computes zdim_{T_0}(n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdim_{T_0}(1) to zdim_{T_0}(n) and the Stirling numbers of the second kind S(n,1) to S(n,n). The resulting algorithms have been implemented in C and we also present results of practical experiments with them. To considerably reduce the running times for computing zdim_{T_0}(n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.

中文翻译：

---

关于覆盖维数的零维有限拓扑空间的算法计数

以覆盖维数dim作为拓扑空间维数的概念，我们首先指定{1,...,n}$上零维T_0-空间的个数zdim_{T_0}(n)和数zdim(n) ) {1,\ldots,n} 上的零维任意拓扑空间，通过两个映射 po 和 P 生成 {1,...,n} 上偏序数 po(n) 和集合 P(n ) 的分区分别为 {1,...,n}。两种映射的算法都存在。假设 po 有一个，我们使用我们的 zdim_{T_0}(n) 规范并修改 P 的一个，这样它计算 zdim_{T_0}(n) 而不是 P(n)。然后，zdim(n) 的规范允许计算从 zdim_{T_0}(1) 到 zdim_{T_0}(n) 和第二类斯特林数 S(n,1) 到 S(n,n) 的这个数. 由此产生的算法已在 C 中实现，我们还提供了实际实验的结果。为了显着减少计算 zdim_{T_0}(n) 的运行时间，我们还使用 OpenMP 库描述了一种回溯方法及其在 C 中的并行实现。