Pre-Hausdorffness and Hausdorffness in Quantale-Valued Gauge Spaces

In this paper, we characterize each of T0, T1, Pre-Hausdorff and Hausdorff separation properties for the category L-GS of quantale-valued gauge spaces and L-gauge morphisms. Moreover, we investigate how these concepts are related to each other in this category. We show that T0, T1 and T2 are equivalent in the realm of Pre-Hausdorff quantale-valued gauge spaces. Finally, we compare our results with the ones in some other categories.