Geodesic
The product of two ordinals is hereditarily dually discrete
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 99-104.

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In Dually discrete spaces, Topology Appl. 155 (2008), 1420--1425, Alas et. al. proved that ordinals are hereditarily dually discrete and asked whether the product of two ordinals has the same property. In Products of certain dually discrete spaces, Topology Appl. 156 (2009), 2832--2837, Peng proved a number of partial results and left open the question of whether the product of two stationary subsets of $\omega_1$ is dually discrete. We answer the first question affirmatively and as a consequence also give a positive answer to the second.
Classification : 54D99, 54F05
Keywords: dually discrete spaces; stationary subsets; ordinal spaces 
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Gaspar-Arreola, M.Á.; Hernández-Hernández, F. The product of two ordinals is hereditarily dually discrete. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 99-104. http://geodesic.mathdoc.fr/item/CMUC_2012__53_1_a6/