Abstract: In the theory of the best approximations the properties (C-K),K=Ⅰ,Ⅱ, Ⅲ play a very significant role (see[1]). The purpose of this paper is to give their interesting characterizations. Further we obtain that if a Banach space X has the property (C-Ⅲ) (resp.(C-Ⅱ)) then for any norming set A of X, the σ(X, A) topology and the weak topology (resp.norm topology) coincide on the unit sphere of X and X is nearly very rotund (resp. nealy strongly rotund).

Key words: Kadec property, Norming set, Near very rotundity, Near stong rotundity, Property (C-K)