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.