Mild normality of finite products of subspaces of ω1

It is known that products of arbitrary many ordinals are mildly normal [L. Kalantan, P.J. Szeptycki, κ-normality and products of ordinals, Topology Appl. 123 (2002) 537–545] and products of two subspaces of ordinals are also mildly normal [L. Kalantan, N. Kemoto, Mild normality in products of ordinals, Houston J. Math. 29 (2003) 937–947]. It was asked if products of arbitrary many subspaces of ordinals are mildly normal. In this paper, we characterize the mild normality of products of finitely many subspaces of ω1. Using this characterization, we show that there exist 3 subspaces of ω1 whose product is not mildly normal.