Mild normality of finite products of subspaces of

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 ω 1 . Using this characterization, we show that there exist 3 subspaces of ω 1 whose product is not mildly normal.