The hyperspaces C(p,X)

Let C(X) denote the hyperspace of subcontinua of a continuum X. For p E X, define the hyperspaces C(p, X) = {A is an element of C(X) : p is an element of A} and K(X) = {C(p,X) : p is an element of X}. Let I denote the unit interval. The class of continua X for which K(X) coincides with K(I) (the class of the so-called arc-similar continua) is characterized as the class of continua having two end-points and arcs as proper nondegenerate subcontinua. Other classes of continua are characterized as well, in terms of the hyperspaces K(X).