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For a space with asymmetric seminorm, the asymmetric closure seminorm is defined as the Minkowski functional of the -closure of the ball . It is shown that an asymmetric normed space is Hausdorff if and only if the closure seminorm is an asymmetric norm on this space. It is also shown that , where . Approximatively compact sets in asymmetric normed -spaces are studied. [ABSTRACT FROM AUTHOR]

For a Hausdorff non-compact space X, relationships between closure-type properties of the hyperspace (Λ,τ+Δ) and covering properties of that of X have been studied. We then investigate selective separability and some variations of this property. Finally supertightness of (Λ,τ+Δ) has been studied. [ABSTRACT FROM AUTHOR]

In this paper, we first give a topological representation of some algebraic lattices of ideals of C(X). Next, we apply these results and prove that a space X is normal if and only if the lattice of closed fixed ideals of C(X) is a sublattice of the lattice of ideals of C(X). It is proved that if two rings C(X) and C(Y) are isomorphic, then two lattices Z∘[X] and Z∘[Y] are isomorphic. We conclude that two rings C*(X) and C*(Y) are isomorphic if and only if two lattices Z[βX] and Z[βY] are isomorphic. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a simple undirected graph. A block of G is a maximal connected subgraph of G that contains no cut-vertices [11]. The family of vertex sets of blocks of G generates a unique topology. In this paper, we formally define the topology generated by the family of blocks in a graph called the block topological space. Moreover, we characterize and describe some special attributes of the block topological space. Finally, we associate a corresponding graph from a given block topological space by defining the block topological graph. [ABSTRACT FROM AUTHOR]

Chipman contended, in stark contrast to the conventional view, that, utility is not a real number but a vector, and that it is inherently lexicographic in nature. On the other hand, in recent years continuous multi-utility representations of a preorder on a topological space, which proved to be the best kind of continuous representation, have been deeply studied. In this paper, we first state a general result, which guarantees, for every preordered topological space, the existence of a lexicographic order-embedding of the Chipman type. Then, we combine the Chipman approach and the continuous multi-utility approach, by stating a theorem that guarantees, under certain general conditions, the coexistence of these two kinds of continuous representations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and give preliminary investigation of ∂-partial-locally compact spaces. Locally compactness and ∂-partial-locally compactness are independent of each other. Every locally compact Hausdorff space is ∂-partial-locally compact. But the converse is not true even though it be Hausdorff. ∂-partial-locally compactness is a topological property. ∂-partial-locally compactness is not preserved by the product topology.

Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces. [ABSTRACT FROM AUTHOR]

The main result of this paper is that, under PFA, for every regular space X with F(X) = \omega we have |X| \le w(X)^\omega; in particular, w(X) \le \mathfrak {c} implies |X| \le \mathfrak {c}. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. In fact, this space X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X intersect, which is much stronger than F(X) = \omega. Moreover, any non-empty open set in X also has size 2^\mathfrak {c}, and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points. [ABSTRACT FROM AUTHOR]

Chipman contended, in stark contrast to the conventional view, that, utility is not a real number but a vector, and that it is inherently lexicographic in nature. On the other hand, in recent years continuous multi-utility representations of a preorder on a topological space, which proved to be the best kind of continuous representation, have been deeply studied. In this paper, we first state a general result, which guarantees, for every preordered topological space, the existence of a lexicographic order-embedding of the Chipman type. Then, we combine the Chipman approach and the continuous multi-utility approach, by stating a theorem that guarantees, under certain general conditions, the coexistence of these two kinds of continuous representations.

In this paper, the notions of L -ideal and L -filter, where L is a complete lattice, are introduced in hoop algebras, and their fundamental properties are investigated. Also, (weak) prime L -ideals, minimal prime L -ideals, and maximal L -ideals are introduced, and in addition to basic properties, the relationships among them are investigated. Particularly, it is proved that the lattices of L -ideals and L -filters are order isomorphic. Next, the Zariski topology induced by weak prime L -ideals is studied; it is proved that it is a compact T 0 -space that is connected, under suitable conditions. Moreover, the (minimal) prime L -ideal space and maximal L -ideal space are studied. Furthermore, a homeomorphism between prime ideal spaces and certain subclass of prime L -ideal spaces is obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a type of compactness, defined in terms of a grill G in a neutrosophic topological space, call it neu-trosophic αG - compact space with the help of neutrosophic α - open cover and neutrosophic G - cover. We would investigate some of its basic properties and characterization theorems in neutrosophic topological space with a grill G. We would define the notion of neutrosophic α - quasi Hausdorff closed space. Then we would establish the relation between neutrosophic αG - compact and neutrosophic α - quasi H - closed spaces. We would also introduce and study neutrosophic αG - compact set relative to a neutrosophic topological space X with a grill G. Lastly we would define countably neutrosophic αG - compact space and investigate some of its basic properties and characterization theorems in a neutrosophic topological space X with a grill G. [ABSTRACT FROM AUTHOR]

Introduced the notions of annulets and 𝒩 -filters in stone Almost Distributive Lattices and investigated their properties. Utilized annulets to characterize the 𝒩 -filters. Derived that every proper 𝒩 -filter is the intersection of all 𝒩 -prime filters containing it and also proved that the set ℱ𝒩 (L) of all 𝒩 -filters is isomorphic to the class ConE(L) of all G-extentions of L. Given some topological properties of the space of all 𝒩 -prime filters. Derived a necessary and sufficient condition for the space of all 𝒩 -prime filters to be a Hausdorff space.

In order to characterize the multiplier in the space of low-dimensional vector-valued functions, this paper derive the expression theorems of bounded operators from C()(Q, H) to H, from C 0 (Q, H) to C, and fromC()(Q) to H. This paper extend the Riesz representation theorem of bounded functional on continuous function spaces over closed intervals to locally compact Hausdorff spaces, and obtain the Riesz representation theorem of vector-valued integrals over Banach spaces whose domain is Hausdorff space and range is Banach space. The Q in this article represents a locally compact Hausdorff space. [ABSTRACT FROM AUTHOR]

In this paper, by considering the notion of effect algebra and by using of a new ideal in an effect algebra E, we construct a topology τ on E, and we show that (E, τ) is a topological effect algebra. Then we obtain some conditions under which that (E, τ) is a Hausdorff space. Also, we obtain some results about connected components of this topological space, and we construct a quotient topological effect algebra. [ABSTRACT FROM AUTHOR]

The notion of boosters and β-filters in stone Almost Distributive Lattices are introduced and their properties are studied, utilizing boosters to characterize the β-filters. It has been derived that every proper β-filter is the intersection of all prime β-filters containing it, and it has also been proved that the set 𝒡β (L) of all β-filters is isomorphic to the set of all ideals of 𝒝0(L). A set of equivalent conditions is derived for 𝒝0(L) to become a relatively complemented Almost Distributive Lattice. Later, some properties of the space of all prime β-filters are derived topologically. Finally, necessary and sufficient conditions are derived for the space of all prime β-filters to be a Hausdorff space.

In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point $p$ in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various ${\rm Pre}T_2$, $T_i$, $i=0,1,2,3$, structures at a point $p$ are investigated. Finally, we examine the relationships between the generalized separation properties and the separation properties at a point $p$ in this category.

In this paper, we shall generalize the concept of equivariance in statistics to "weak equivariance". Then, we summarize the properties of weakly equivariant estimators and their applications in statistics. At first we characterize the class of all weakly equivariant estimators. Then, we shall consider the concept of cocycles and isovariance, and so we find their connection with weakly equivariant functions. It is natural to restrict attention to the class of weakly equivariant estimator to find minimum risk weakly equivariant estimators. If the group acts in two different ways, we shall find a relation between the minimum risk equivariant and minimum risk weakly equivariant estimator under the old and new group actions. Also we shall introduce a necessary and sufficient condition for the invariance of the loss function under the new action. [ABSTRACT FROM AUTHOR]

The concept of generalized prime D-filters is introduced in distributive lattices. Generalized prime D-filters are characterized in terms of principal filters and ideals. The notion of generalized minimal prime D-filters is introduced in distributive lattices and properties of minimal prime D-filters are then studied with respect to congruences. Some topological properties of the space of all prime D-filters of a distributive lattice are also studied. [ABSTRACT FROM AUTHOR]

Some properties of prime D-filters of distributive lattices are studied with respect to the direct products and set-theoretic intersection. Properties of minimal prime D-filters of distributive lattices are studied. Some topological properties of the space of all minimal prime D-filters of a distributive lattice are studied. Finally, it is showed that the space of all minimal prime D-filters of a distributive lattice is a Hausdorff space. [ABSTRACT FROM AUTHOR]

The notion of boosters and β-filters in stone Almost Distributive Lattices are introduced and their properties are studied, utilizing boosters to char- acterize the β-filters. It has been derived that every proper β-filter is the intersection of all prime β-filters containing it, and it has also been proved that the set Fβ(L) of all β-filters is isomorphic to the set of all ideals of B0(L). A set of equivalent conditions is derived for B0(L) to become a rela- tively complemented Almost Distributive Lattice. Later, some properties of the space of all prime β-filters are derived topologically. Finally, necessary and sufficient conditions are derived for the space of all prime β-filters to be a Hausdorff space. [ABSTRACT FROM AUTHOR]

In previous papers, various notions of pre-Hausdorff, Hausdorff and regular objects at a point p in a topological category were introduced and compared. The main objective of this paper is to characterize each of these notions of pre-Hausdorff, Hausdorff and regular objects locally in the category of proximity spaces. Furthermore, the relationships that arise among the various PreT2, Ti, i = 0, 1, 2, 3, structures at a point p are investigated. Finally, we examine the relationships between the generalized separation properties and the separation properties at a point p in this category. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a necessary condition for the density of a sum of finitely many algebras in the space of continuous functions. This condition complements the known sufficient condition of Sproston and Straus [22]. In case of two algebras and under a suitable restriction, our necessary condition turns out to be also sufficient. [ABSTRACT FROM AUTHOR]

The conventional extension principle is established on the Euclidean space and defined by considering the minimum or t-norm operator in which the fuzzy sets are usually assumed to be normal. The previous work on generalized extension principle was also based on the normal fuzzy sets. Since the non-normal fuzzy sets occur frequently in practical applications, in this paper, the generalized extension principle based on the non-normal fuzzy sets is established in which the general aggregation operator and Hausforff space are taken into account. [ABSTRACT FROM AUTHOR]

In this paper, we define the notions of sequentially Hausdorff Space and full sequentially Hausdorff space. Also we give the relationships between these notions and Hausdorffness. [ABSTRACT FROM AUTHOR]