The product of two ordinals is hereditarily dually discrete.
The main result is the following theorem: Let \(\mu \) and \(\nu \) be two ordinals (considered with the order topology) and let \(Y\) be an arbitrary subset of the product \(\mu \times \nu \). Then \(Y\) is dually discrete, that is, for any neighborhood assignment \(y\mapsto \phi (y)\), \(y\in Y\), there is a discrete subset \(D\subset Y\) (not necessarily closed) such that the sets \(\phi (y)\), \(y\in D\), cover \(Y\).NEWLINENEWLINEThis result was independently proved by \textit{L.-X. Peng} [Topology Appl. 159, No. 1, 304--307 (2012; Zbl 1234.54034)] using a very similar method.
scientific article; zbMATH DE number 6054290