F. Esmaeeli - Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box ۱۱۵۹, Mashhad, Iran.
K. Mirzavaziri - Department of Computer Science, School of Mathematics, Statistics and Computer Science, University of Tehran, P.O. Box ۱۴۱۵۵۶۶۱۹, Tehran, Iran.
M. Mirzavaziri - Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box ۱۱۵۹, Mashhad, Iran.

‎For a positive integer m‎, ‎a subset of divisors of m is called a \textit{divisor topology on m} if it contains ۱ and m and it is closed under taking \gcd and \rm lcm‎. ‎If m=p_۱\dots p_n is a square free positive integer‎, ‎then a divisor topology m corresponds to a topology on the set [n]=\{۱,۲,\ldots,n\}‎. ‎Giving some facts about divisor topologies‎, ‎we give a recursive formula for the number of divisor topologies on a positive integer‎.

Topology, Divisor topology, Semi-divisor topology