On almost sober spaces
A topological \(T_0\) space \(X\) is called \textit{almost sober} iff each irreducible closed set has a join in the specialization order of \(X\). It is shown that almost sober spaces are closed under retracts, products and saturated subspaces. Other basic properties known from sober spaces are shown not to hold for almost sobriety, for example closed-hereditariness or the stability with respect to forming function spaces with the topology of pointwise convergence. Examples illustrate that almost sobriety of \(X\) is neither necessary nor sufficient for almost sobriety of its Smyth power space (the nonempty saturated compacta of \(X\) equipped with the upper Vietoris topology). The final section proves that the category of almost sober spaces with continuous mappings is not reflective in the category of \(T_0\) spaces and continuous mappings. The famous example of a non-sober Scott space [\textit{P. T. Johnstone}, Lect. Notes Math. 871, 282--283 (1981; Zbl 0469.06002)] functions prominently in this deduction.
scientific article; zbMATH DE number 7434156
scientific article