Pre-Hausdorff and Hausdorff objects in the category of quantale-valued closure spaces

2022

Filomat

In this paper, we study the category of quantale-valued preordered spaces. We show that it is a normalized topological category and give characterization of zero-dimensionality and D-connectedness in the category of quantale-valued preordered spaces. Moreover, we characterize explicitly each of T0, T0, T1, pre-T2, T2 and NT2 quantale-valued preordered spaces. Finally, we examine how these characterization are related to each other and show that the full subcategory Ti(pre-T2(L-Prord)) (i=0,1,2) of pre-T2(L-Prord), and the full subcategory Ti (L-Prord) (i=1,2) of L-Prord are isomorphic.

Section: Discussionmentioning

confidence: 99%

Zero-dimensionality and Hausdorffness in quantale-valued preordered spaces

Özkan

2022

Filomat

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

Malik¹,

Khyzer²,

2022

MATH

<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>

Some topological aspects of interval spaces

Khan

Alsulami

et al. 2022

MATH

<abstract><p>In previous papers, several $ T_{0} $, $ T_{2} $ objects, $ D $-connectedness and zero-dimensionality in topological categories have been introduced and compared. In this paper, we characterize separated objects, $ T_{0} $, $ {\bf{T_{0}}} $, $ T_{1} $, Pre-$ T_{2} $ and several versions of Hausdorff objects in the category of interval spaces and interval-preserving mappings and examine their mutual relationship. Further, we give the characterization of the notion of closedness and $ D $-connectedness in interval spaces and study some of their properties. Finally, we introduce zero-dimensionality in this category and show its relation to $ D $-connectedness.</p></abstract>