About The Set Function Г

The set functionΓis introduced by David P.Bellamy.This paper includes three aspects.The first is the study of some properties of the set functionΓ,giving some results of the additivity,symmetry of the set functionΓ;The second obtains some results on the continuity of set functionΓ,giving that set functionΓis always be upper semicontinuous and under the condition of the continuity of set functionΓ,the family {Γ({x})|x∈X} resembles an upper semicontinuous decomposition;The third presents several applications of set functionΓ,giving a sufficient condition for a continuum that is not contractible.