A note on monotone $D$-spaces

Abstract

A topological space \((X, \tau)\) is a \(D\)-space if for every function \(\varphi\): \(X\rightarrow \tau\) with \(x\in \varphi(x)\) for each \(x\in X\), \(\{\varphi (x): x\in F\}\) covers \(X\) for some closed discrete subset \(F\) of \(X\). The Michael line \(M\), one of the most important elementary examples in general topology, is the Euclidean space \(\mathbb{R}\) isolating the irrationals. In this note we show that (1) the minimal dense linearly ordered extension of \(M\) is hereditarily paracompact, but not monotonically \(D\); (2) the minimal closed linearly ordered extension of \(M\) is monotonically \(D\); (3) if the space \(X\) is a \(D\)-space (resp., a monotone \(D\)-space), then so is its Alexandroff duplicate space \(\mathscr{A}(X)\) and thus \(\mathscr{A}(M)\) is monotonically \(D\).