On C-embeddedness of hyperspaces

Let CL(X) and K(X) denote the hyperspaces of non-empty closed and non-empty compact subsets of X, respectively, with the Vietoris topology. In this paper we show that, given an ordinal number γ, the space K([0,γ)) is C-embedded in CL([0,γ)) if and only if cof(γ)≠ω. Moreover we answer some problems posed by the first author and Jun Terasawa.