The categories $L$\textbf{-Top_0} and $L$\textbf{Sob} as epireflective hulls

Abstract

We show that the category $L{\text{-Top}}_0$ of $T_0$-$L$-topological spaces is the epireflective hull of Sierpinski $L$-topological space in the category $L\text{-Top}$ of $L$-topological spaces and the category $L\text{-Sob}$ of sober $L$-topological spaces is the epireflective hull of Sierpinski $L$-topological space in the category $L{\text{-Top}}_0$.