Covering dimension d by a normal base

S.D. Iliadis introduced the concept of the dimension-like functions of type dim using the notion of a normal base. Since he considered this notion only from the point of view of the existence of universal spaces in different classes of spaces, in the present paper the basic theorems of dimension theory are obtained for these functions. Their relationship with the relative dimensions of A. Chigogidze and the uniform dimensions of M. Charalambous are shown.