Resolutions and dimension

In the present paper we give a partial answer to the question: Let p : X -> X = {;X_a , f_ab , A}; be a resolution with dim X_a < n. Is it true that dim X < n and dam (lim X) < n. The main result of Section One is a characterization of a base of X. An important property of a resolution gives 1.8. Section Two is devoted to the mappings p_a:X-> X_a. The closedness of p_a is proved in 2.1. Section Three is the main section. The positive answer to the preceeding question is given for dim and dim_f. It is proved that a normal space X is countably compact with dim X < n iff there is a resolution p : X -> X = {; X_a, f_ab, A) such that X_a, a in A, are compact metric spaces with dim X_a < n. Section Four contains some results concerning the dimension Ind and ind.