On Nagataʹs star-index ∗k(X)

In [Questions Answers Gen. Topology 2 (1984) 2–13], Nagata introduced a new function ∗k(X) for a normal space X and k∈N by modifying the function Δk(X) due to Bruijning and Nagata [Pacific J. Math. 80 (1979) 1–8]. We shall determine the value of ∗k(X) exactly. This answers a question of Nagata [Questions Answers Gen. Topology 2 (1984) 2–13, Question 5].