For an increasing sequence xn, n = 1, 2, . . . , of positive integers define the block sequence Xn = (x1/xn, . . . , xn/xn). We study the set G(Xn) of all distribution functions of Xn, n = 1, 2, . . . . We find a special xn such that G(Xn) is not connected and we give some criterions for connectivity of G(Xn). We also give an xn such that G(Xn) contains one-step distribution function with step 1 in 1 but does not contain one-step distribution function with step 1 in 0. We prove that if G(Xn) is constituted by one-step distribution functions, at least two different, then it contains distribution functions with steps in 0 and 1.