Dual properties of subspaces in products of ordinals (Q989112)

A neighbourhood assignment on a space \(X\) is a function \(N\) from \(X\) to the topology of \(X\) such that \(x\in N(x)\) for any \(x\in X\). Given a topological property \(\mathcal P\), say that a space \(X\) is \textit{dually \(\mathcal P\)} if, for any neighbourhood assignment \(N\) on the space \(X\), there exists a subspace \(K\subset X\) such that \(K\) has \(\mathcal P\) and \(\bigcup\{N(x):x\in K\}=X\). For any space \(X\) let \(I(X)\) be the set of isolated points of \(X\); if \(X\) is discrete or \(X\setminus I(X)\) is discrete then \(X\) is called scattered of rank \(\leq2\). It is proved that for any ordinals \(\mu\) and \(\nu\), if \(X \subset \mu\times \nu\) then \(X\) is dually scattered of rank \(\leq2\); if, additionally, \(X\) has countable extent then \(X\) is dually discrete.