Previously, several characterization of local Pre-Hausdorffness and $D$-connectedness have been examined in distinct topological categories. In this paper, we give the characterization of local $T_0$ (resp. local $T_1$) $\mathcal{L}$-valued closure spaces, examine how their mutual relationship. Furthermore, we give the characterization of a closed point and $D$-connectedness in $\mathcal{L}$-valued closure spaces and examine their relations with local $T_0$ and local $T_1$ objects. Finally, we examine the characterization of local Pre-Hausdorff and local Hausdorff $\mathcal{L}$-valued closure spaces and study their relationship with generic Hausdorff objects and $D$-connectedness.

Previously, several notions of $T_0$ and $T_1$ objects have been studied and examined in various topological categories. In this paper, we characterize each of $T_0$ and $T_1$ objects in the categories of several types of bounded uniform filter spaces and examine their mutual relations, and compare that with the usual ones. Moreover, it is shown that under $T_0$ (resp. $T_1$) condition, the category of preuniform (resp. semiuniform) convergence spaces and the category of bornological (resp. symmetric) bounded uniform filter spaces are isomorphic. Finally, it is proved that the category of each of $T_0$ (resp. $T_1$) bounded uniform filter space are quotient reflective subcategories of the category of bounded uniform filter spaces.