The product of two ordinals is hereditarily dually discrete
A topological space \((X,\tau)\) is called dually discrete (respectively, a \(D\)-space) if every mapping \(U: X\to\tau\) with \(x\in U(x)\) for each \(x\in X\) has a discrete (respectively, closed discrete) kernel, where a subset \(A\) of \(X\) is a kernel of \(U\) if \(\bigcup\{U(x)\mid x\in A\}= X\). \(D\)-spaces were introduced by \textit{E. K. van Douwen} in a joint paper with \textit{W. F. Pfeffer} [Pac. J. Math. 81, 371--377 (1979; Zbl 0409.54011)]. A recent survey on \(D\)-spaces by \textit{G. Gruenhage} can be found in [A survey of \(D\)-spaces. Babinkostova, L. (ed.) et al., Set theory and its applications. Annual Boise extravaganza in set theory, Boise, ID, USA, 1995--2010. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 533, 13--28 (2011; Zbl 1217.54025)]. Dually discrete spaces were introduced by \textit{J. van Mill}, \textit{V. V. Tkachuk} and \textit{R. G. Wilson} in [Topology Appl. 154, No. 10, 2127--2134 (2007; Zbl 1131.54022)] and subsequently studied by \textit{O. T. Alas}, \textit{R. Z. Buzyakova}, \textit{L. R. Junqueira}, \textit{V. V. Tkachuk}, \textit{R. G. Wilson} and the present author. In [ibid. 155, No. 13, 1420--1425 (2008; Zbl 1169.54010)], \textit{O. T. Alas}, \textit{L. R. Junqueira} and \textit{R. G. Wilson} have shown that every ordinal with its natural order topology is hereditarily dually discrete. As an answer to one of their questions it is shown here that, more generally, every product of two ordinals is hereditarily dually discrete. It follows that, if \(\mu\), \(\nu\) are ordinals and \(X\) is a subspace of \(\mu\times\nu\), then the following are equivalent: (i) \(X\) has countable spread; (ii) \(X\) is perfectly normal and has countable extent; and (iii) \(X\) is hereditarily a Lindelöf \(D\)-space.
scientific article; zbMATH DE number 5986567