Let C be the collection of continuous self-maps of the unit interval I = [ 0 , 1 ] to itself. For f ∈ C and x ∈ I , let ω ( x , f ) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q ( x , f ) to be the intersection of all the asymptotically stable sets of f containing ω ( x , f ) . We show that Q ( x , f ) tells us quite a bit about the stability of ω ( x , f ) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q ( x , f ) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω ( x , g ) . We also study the structure of the map Q taking ( x , f ) ∈ I × C to Q ( x , f ) . We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I × C . We also consider the map Q f : I → K given by x ↦ Q ( x , f ) , and find that this map is continuous if and only if it is a constant map; that is, only when the set Q ( f ) = { Q ( x , f ) : x ∈ I } is a singleton.