For a Tychonoff space X, we denote by Ck(X) the space of all real-valued continuous functions on X with the compact-open topology. A subset A ⊂ X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A. In this paper, the following properties for Ck(X) are considered. S1(S,S)⇒ S f in(S,S)⇒ S f in(S,D)⇐ S1(S,D) ⇑ ⇑ ⇑ ⇑ S1(D,S)⇒ S f in(D,S)⇒ S f in(D,D)⇐ S1(D,D) For example, a space Ck(X) satisfies S1(S,D) (resp., S f in(S,D)) if whenever (Sn : n ∈N) is a sequence of sequentially dense subsets of Ck(X), one can take points fn ∈ Sn (resp., finite Fn ⊂ Sn) such that { fn : n ∈ N} (resp., ⋃{Fn : n ∈N}) is dense in Ck(X). Other properties are defined similarly. In [22], we obtained characterizations these selection properties for Cp(X). In this paper, we give characterizations for Ck(X).