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

Abstract

Taking the covering dimension dim as notion for the dimension of a topological space, we first specify the number zdimT0(n) of zero-dimensional T0-spaces on {1,…,n} and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,…,n} by means of two mappings po and P that yield the 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 zdimT0(n) and modify one for P in such a way that it computes zdimT0(n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdimT0(1) to zdimT0(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 zdimT0(n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.