 There is a corresponding inverse system { "π"0( "X" \ "K" ) }, where "π"0("Y") denotes the set of connected components of a space "Y", and each inclusion map "Y" → "Z" induces a function "π"0("Y") → "π"0("Z").Then set of ends of "X" is defined to be the inverse limit of this inverse system.Under this definition, the set of ends is a functor from the category of topological spaces, where morphisms are only "proper" continuous maps, to the category of sets.