Group invariants such as homotopy groups, and cardinal invariants such as weights and density character are very often used in the problem of classifying topological spaces. Some interesting and useful ordinal invariants introduced and investigated in the recent past have proved equally significant in this direction. The k-order and sequential order are two of them. In this paper, two natural open questions concerning them are answered fully. The main result of the paper is the theorem which assetts, for each ordinal α, the existence of a nice k-space of k-order α. © 1975.