Local Pre-Hausdorffness and D-connectedness in $\mathcal{L}$-valued closure spaces

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.