A note on the rank of diagonals

For each natural number k ⩾ 4 , we construct a Tychonoff space with a rank k-diagonal but without a rank ( k + 1 ) -diagonal. This example proves a conjecture on rank of diagonal given by A.V. Arhangelʹskii and R.Z. Buzyakova (2006) in [1] and answers some questions raised by them in the same paper.