Compactifications whose cone can be embedded into their hyperspace of subcontinua

Given a metric continuum X  , let C(X)C(X) be the hyperspace of subcontinua of X   and Cone(X)Cone(X) the topological cone of X. We say that a continuum X   is cone-embeddable in C(X)C(X) provided that there is an embedding h   from Cone(X)Cone(X) into C(X)C(X) such that h(x,0)={x}h(x,0)={x} for each x in X. In this paper, we present some results concerning compactifications X   of rays, union of two rays, and real lines which are cone-embeddable in C(X)C(X).