In this paper, we improve almost all main results in ref. [5]. Especially, Theorem 1, Theorem 9, Theorem 10 and Theorem 12 improve Arhangelskii's inequality, Sapirouskii's inequality, Hajnal-Juhasz inequality and some better inequalities, respectively. Theorem 8 and Theorem 9 expose some new essential character about cardinal function sqL(X).