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Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension

by   Rudolf Berghammer, et al.

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,...,nand the number zdim(n) of zero-dimensional arbitrary topological spaces on 1,...,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.


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