On the number of topologies on a finite set

Kızmaz, Muhammet Yasir
We denote the number of distinct topologies which can be defined on a set X with n elements by T(n). Similarly, T-0(n) denotes the number of distinct T-0 topologies on the set X. In the present paper, we prove that for any prime p, T(p(k)) k+ 1 (mod p), and that for each natural number n there exists a unique k such that T(p + n) k (mod p). We calculate k for n = 0, 1, 2, 3, 4. We give an alternative proof for a result of Z. I. Borevich to the effect that T-0(p + n) T-0(n + 1) (mod p).

Citation Formats
M. Y. Kızmaz, “On the number of topologies on a finite set,” ALGEBRA & DISCRETE MATHEMATICS, vol. 27, no. 1, pp. 50–57, 2019, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/54206.