Geodesic
Bruijning--Nagata and Hashimoto--Hattori characteristics of covering dimension revisited
Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 353-361.

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New dimension functions for topological spaces are introduced in the spirit of Nagata's approach. Expressions for the new functions in terms of covering dimension include the Bruijning–Nagata and Hashimoto–Hattori formulas. 
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S. A. Bogatyi; A. N. Karpov. Bruijning--Nagata and Hashimoto--Hattori characteristics of covering dimension revisited. Matematičeskie zametki, Tome 79 (2006) no. 3, pp. 353-361. http://geodesic.mathdoc.fr/item/MZM_2006_79_3_a2/

[1] Bruijning J., Nagata J., “A characterization of covering dimension by use of $\Delta_k(X)$”, Pacific J. Math., 80:1 (1979), 1–8 | MR | Zbl

[2] Nagata J., “Some questions, II”, Questions Answers Gen. Topology, 2 (1984), 2–13 | MR | Zbl

[3] Hashimoto K., Hattori Y., “On Nagata's star-index $*_k(X)$”, Topol. Appl., 122 (2002), 201–204 | DOI | MR | Zbl