This work is about symmetric numerical semigroups. The symmetric numerical semigroups are very important in semigroup theory. They play an important role especially in algebraic geometry, coding theory and other areas of algebra. Here, we examine Arf closure of S_q symmetric numerical semigroup, and give some relations between S_q and Arf closure S_q such that where p is a prime number and for 

determine number, symmetric numerical semigroups, Arf closure, gaps.