Let f be a continuous map of the circle S 1 into itself. And let R ( f ) , Λ ( f ) , Γ ( f ) , and Ω ( f ) denote the set of recurrent points, ω -limit points, γ -limit points, and nonwandering points of f , respectively. In this paper, we show that each point of Ω ( f ) \ R ( f ) ¯ is one-side isolated, and prove that

(1) Ω ( f ) \ Γ ( f ) is countable and

(2) Λ ( f ) \ Γ ( f ) and R ( f ) ¯ \ Γ ( f ) are either empty or countably infinite.