In this paper, an explicit characterization of the separation properties T0
and T1 at a point p is given in the topological category of proximity spaces.
Furthermore, the (strongly) closed and (strongly) open subobjects of an
object are characterized in the category of proximity spaces and also the
characterization of each of the various notions of the connected objects in
this category are given.
In this paper, an explicit characterization of the separation properties for
T0, T1, PreT2 (pre-Hausdorff) and T2 (Hausdorff) is given in the topological
category of proximity spaces. Moreover, specific relationships that arise
among the various Ti,i = 0,1,2 and PreT2 structures are examined in this
category. Finally, we investigate the relationships among generalized
separation properties for Ti,i = 0,1,2 and PreT2 (in our sense),
separation properties at a point p and separation properties
for Ti, i = 0,1,2 in the usual sense in this category.
In this paper, we characterize local T0 and T1 quantale-valued gauge spaces, show how these concepts are related to each other and apply them to L-approach distance spaces and L-approach system spaces. Furthermore, we give the characterization of a closed point and D-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each other.
We characterize the separation properties T0 and T1 at a point p in the
category of quasi-proximity spaces. Moreover, the (strongly) closed and
(strongly) open subobjects of an object, and each of the various notions of
connected and compact objects are characterized in this topological
category.
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.
In this paper, we characterize explicitly the separation properties $T_0$ and $T_1$ at a point p in the topological category of quantale-valued preordered spaces and investigate how these characterizations are related. Moreover, we prove that local $T_0$ and $T_1$ quantale-valued preordered spaces are hereditary and productive.
In this paper, the notion of single-valued neutrosophic proximity spaces which is a generalisation of fuzzy proximity spaces [Katsaras AK. Fuzzy proximity spaces. Anal and Appl. 1979;68(1):100-110.] and intuitionistic fuzzy proximity spaces [Lee SJ, Lee EP. Intuitionistic fuzzy proximity spaces. Int J Math Math Sci. 2004;49:2617-2628 was introduced and some of their properties were investigated. Then, it was shown that a single-valued neutrosophic proximity on a set X induced a single-valued neutrosophic topology on X. Furthermore, the existence of initial single-valued neutrosophic proximity structure is proved. Finally, based on this fact, the product of single-valued neutrosophic proximity spaces was introduced.
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