On the number of topologies on a finite set


Kizmaz M. Y.

ALGEBRA & DISCRETE MATHEMATICS, cilt.27, sa.1, ss.50-57, 2019 (ESCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 27 Konu: 1
  • Basım Tarihi: 2019
  • Dergi Adı: ALGEBRA & DISCRETE MATHEMATICS
  • Sayfa Sayıları: ss.50-57

Özet

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).