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

Abstract

<abstract><p>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.</p></abstract>