Unions of left-separated spaces (Q1705198)

A topological space \(X\) is left-separated if there exists a well ordering of \(X = \{x_{\alpha}: \alpha < \kappa\}\) so that each initial segment is closed in \(X\), cf. \textit{A. Hajnal} and \textit{I. Juhász} [Nederl. Akad. Wet., Proc., Ser. A 70, 343--356 (1967; Zbl 0163.17204)]. It is known that the union of two left-separated spaces need not be left-separated. The authors investigate under which conditions the union of left-separated spaces \(X\) and \(Y\) is left-separated. The main result states that it is the case if ord\(_{\ell}(X) \geq \omega_1\), ord\(_{\ell}(Y) = \omega_1\) and \(X\cup Y\) is locally countable. Here ord\(_{\ell}(X)\) is the smallest \(\kappa\) witnessing left-separation of \(X\). Six open problems are provided.