Kan extendable subcategories and fibrewise topology

The present paper studies certain low-level separation axioms of a topological space, denoted by A(X), induced by a geometric AC-complex X. After proving that whereas A(X) is an Alexandroff space satisfying the semi-T1 2 -separation axiom, we observe that it does neither satisfy the pre T1 2 -separation axiom nor is a Hausdorff space. These are main motivations of the present work. Although not every A(X) is a semi-T1 space, after proceeding with an edge to edge tiling (or a face to face crystallization) of Rn, n ? N, denoted by T(Rn) as an AC complex, we prove that A(T(Rn)) is a semi-T1 space. Furthermore, we prove that A(En), induced by an nD Cartesian AC complex Cn = (En,N,dim), is also a semi-T1 space, n ? N. The paper deals with AC-complexes with the locally finite (LF-, for brevity) property, which can be used in the fields of pure and applied mathematics as well as digital topology, computational topology, and digital geometry.

Here we have studied the ideas of $g^*$-closed sets, $g\wedge_\tau $-sets and $\lambda^*$-closed sets and investigate some of their properties in the spaces of A. D. Alexandroff [1]. We have also studied some separation axioms like $ T_\frac{\omega}{4}, T_\frac{3\omega}{8}, T_\omega $ in Alexandroff spaces and also have introduced a new separation axiom namely $ T_\frac{5\omega}{8} $ axiom in this space.

We develop the separation axioms in fuzzy sequential topological spaces and establish some results related to those axioms. Notions of various separation axioms in fuzzy sequential topological spaces are introduced and investigated the relations among them. Dependency of a component on another component of a fuzzy sequential topology plays the main role in this paper.

Comparing Cartesian closed categories of (core) compactly generated spaces

On extension functions for image space with different separation axioms

The purpose of this article is to introduce new types of limit points and separation axioms on supra topological spaces by using supra $\\beta$-open sets. We explore some characterizations and explain the relationships between them with the help of examples. Also, we study many features of them and give sufficient conditions for some equivalent relations between them.

The theory of knowledge spaces (KST) which is regarded as a mathematical framework for the assessment of knowledge and advices for further learning. Now the theory of knowledge spaces has many applications in education. From the topological point of view, we discuss the language of the theory of knowledge spaces by the axioms of separation and the accumulation points of pre-topology respectively, which establishes some relations between topological spaces and knowledge spaces; in particular, we show that the language of the regularity of pre-topology in knowledge spaces and give a characterization for knowledge spaces by inner fringe of knowledge states. Moreover, we study the relations of Alexandroff spaces and quasi ordinal spaces; then we give an application of the density of pre-topological spaces in primary items for knowledge spaces, which shows that one person in order to master an item, she or he must master some necessary items. In particular, we give a characterization of a skill multimap such that the delineated knowledge structure is a knowledge space, which gives an answer to a problem in [14] or [18] whenever each item with finitely many competencies; further, we give an algorithm to find the set of atom primary items for any finite knowledge space.

Elementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed.

We characterize the existence of a real continuous order-preserving function on a topological preordered space, under the hypotheses that the topological space is normal and the preorder satisfies a strong continuity assumption, called IC-continuity. Under the same continuity assumption concerning the preorder, we present a sufficient condition for the existence of a continuous order-preserving function in case that the topological space is completely regular.

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

A new category is described, which generalizes a select variety of categories having smooth mappings as their class of morphisms, those like the C∞ manifolds and C∞ mappings between them. Categorical embeddings are produced to justify this claim of generalization. Theorems concerning the equivalence of smooth and continuous versions of different separation axioms are proved following the categorical discussion. These are followed by a generalization of Whitney's approximation theorem, a smooth version of the Tietze extension theorem, and a sufficient condition to guarantee that the connected components of these spaces are smoothly path-connected.

In this paper, we investigate the Zariski topology on the prime spectrum of commutative Krasner hyperrings and explore its interplay with the underlying algebraic structure. We characterize the topological properties of the spectrum, such as connectedness, irreducibility, compactness, and separation axioms, and provide necessary and sufficient conditions for each. Notably, we show that the spectrum is irreducible if and only if the nilradical is a prime hyperideal, and it is connected precisely when the hyperring is not a non-trivial product. We also study functorial behavior of the Zariski topology in the category of hyperrings and analyze its correspondence with classical ring theory via the fundamental relation γ∗. Furthermore, we define a topology on the space of prime strongly regular relations and establish a homeomorphism with a subspace of the classical spectrum. These results contribute to the categorical and topological foundations necessary for developing a sheaf-theoretic framework in the context of hyperrings.

In recent years, some generalized structures of topology were introduced. Supra topology was one the most important of those generalizations. To contribute in this orientation, we devoted this work to studying limit points and separation axioms on supra topological spaces by using supra $b$-open sets. We define them in a similar way of their counterparts on topological spaces. In general, we demonstrate their main properties and investigate the sufficient conditions for some equivalent relations between them. Some novel and interesting examples are provided.

Several notions on soft topology are studied and their basic properties are investigated by using the concept of soft pre open sets and soft pre closure operator which are derived from the basics of soft set theory established by Molodtsov [1]. In this paper we introduce some soft separation axioms called Soft pre R0 and soft pre R1 in soft topological spaces which are defined over an initial universe with a fixed set of parameters. Many characterizations and properties of these spaces have been demonstrated. Necessary and sufficient conditions for a soft topological space to be a soft pre Ri space for i = 0, 1 were also presented. Furthermore, the concept of functions with soft pre closed graph and soft pre cluster set are defined. Many results on these two concepts are proved. Also, it is proved that a function has a soft pre closed graph if and only if its soft pre cluster set is degenerate.

One of the important subclasses of abelian paratopological groups is called the free abelian paratopological group on a topological space. It was introduced in 2003 by Remaguara, Sanchis, and Tkachenko. In this paper, we introduce a single submonoid of the free abelian paratopological group on Alexandroff space, then we prove that this submonoid is a base at the identity element of the free topology of the abelian group. The main result of this paper is to give applications of this submonoid such as studying separation axioms, compactness, and other properties of the free topology on the abelian group. Keywords: free group, Abelian paratopological group, free abelian paratopological group, Alexandroff space, neighborhood base at an identity.