\(\alpha\)-sober spaces via the orthogonal closure operator (Q1364912)

For each ordinal \(\alpha\geq 2\), consider the space \(S(\alpha)\) of all ordinals less than \(\alpha\), equipped with the upper topology. The author provides various descriptions of the members (called \(\alpha\)-sober space) of the reflective hull \({\mathbf {Sob}}(\alpha)\) of \(S(\alpha)\) in {\textbf{Top}} [= the epireflective hull of \(S(\alpha)\) in \({\mathbf {Top}}_0\)], using as main tool orthogonal closure operators. In particular for finite \(\alpha\), the \(\alpha\)-sober spaces are the sober spaces. Moreover, it is shown that \(\alpha<\beta\) implies \(\text{Sob}(\alpha)\subset \text{Sob}(\beta)\), and the latter inclusion is proper if and only if there is some infinite regular cardinal \(\lambda\) with \(\alpha\leq \lambda\leq \beta\). Thus sobriety becomes the first step in a rather natural large hierarchy of properties of \(T_0\)-spaces.