Mappings and spaces defined by the function epsilon

Abstract The function e X assigns to each point of a given continuum X the closure of the family of all continua that contain x in their interior. We define the class S ( e ) of continua for which the function e X is continuous. On the other hand, we consider some natural diagram involving the function e X and commutativity of this diagram defines a class of mappings M ( e ) . We investigate classes S ( e ) and M ( e ) , and relations between them.