On fuzzy sobriety (Q1298346)

A \(T_0\) topological space \(X\) is called sober iff every irreducible non-empty closed set of \(X\) is of the form \(\overline x\) for a (unique) \(x\in X\). It is known that the category of sober spaces is the epireflective hull of the Sierpinski space 2 in the category of \(T_0\) spaces. \textit{S. E. Rodabaugh} [Fuzzy Sets Syst. 40, No. 2, 297-345 (1991; Zbl 0733.54003)], \textit{D. Zhang} and \textit{Y. Liu} [ibid. 56, No. 2, 215-227 (1993; Zbl 0787.54011); 76, No. 2, 259-270 (1995; Zbl 0852.54008)] generalized the idea of sobriety to fuzzy topological spaces and stratified fuzzy topological spaces respectively. In this paper the author proved that the category of (stratified) fuzzy sober spaces is the epireflective hull of the (stratified resp.) fuzzy Sierpinski space \((I,\Delta)\) \(((I,\Delta^c))\) in the category of (stratified resp.) \(T_0\) fuzzy spaces, where \(\Delta= \{0,1,id\}\) and \(\Delta^c\) is the stratification of \(\Delta\).