Alexandroff Spaces Research Articles

T_0 functional Alexandroff topologies are partial metrizable

If f : X → X is a function, the associated functional Alexandroff topology on X is the topology whose closed sets are { A ⊆ X : f ( A ) ⊆ A } . We prove that every functional Alexandroff topology is pseudopartial metrizable and every T0 functional Alexandroff topology is partial metrizable.

Necessary or sufficient condition for Alexandroff topological spaces to be cordial graphic

In this paper, we explore the property of being a cordial graphic and establish that it corresponds to an Alexandroff topological space. We analyze how the characteristics of cordial graphs align with the principles of Alexandroff topology and provide insights into their topological structure.

Crypto-preorders, topological relations, information and logic

ABSTRACT As is well known, any preorder R on a set U induces an Alexandrov topology on U . In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on U even if R is not a preorder. If this is the case, then we call R a crypto-preorder. The paper studies the conditions under which a relation R is a crypto-preorder and how to transform a crypto-preorder into a proper preorder.

مونويدات جزئية من الزمر الباراتبولوجية الآبيلية

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.

Avoidance Spectrum of Alexandroff Spaces

In this paper we prove that every T0 Alexandroff topological space (X, τ ) is homeomorphic to the avoidance of a subspace of (Spec(Λ), τZ), where Spec(Λ) denotes the prime spectrum of a semi-ring Λ induced by τ and τZ is the Zariski topology. We also prove that (Spec(Λ), τZ) is an Alexandroff space if and only if Λ satisfies the Gilmer property.

Basic concepts of complete residuated lattice-valued fuzzy mathematical morphology

The aim of this paper is to present basic concepts of lattice-valued fuzzy mathematical morphology. We use a complete residuated lattice as the codomain of fuzzy sets, a pair of fuzzy powerset operators, called the fuzzy erosion operator and the fuzzy dilation operator, is defined and their properties and relationships are studied. The pair of two operators forms a Galois adjunction and so that the induced fuzzy opening operator and fuzzy closing are an interior operator and a closure operator respectively. It is shown that the dilation stable sets and the erosion stable sets are equivalent, which form a fuzzy Alexandrov topology.

The proper homotopy of Alexandroff spaces

This paper introduces the study of the infinity in the class of Alexandroff spaces by establishing the proper category of locally finite Alexandroff spaces. We observe that the McCord and Clader theorems are available in this setting. The paper can be considered as the starting point of a seemingly new research: relating the combinatorics and topology of the proper category of Alexandroff spaces to the proper invariants used in the geometric topology of non-compact polyhedra and manifolds.

On Protected Quasi-Metrics

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations.

Construction of an Alexandroff topology on the edge set of an undirected graph

Construction of an Alexandroff topology on the edge set of an undirected graph

Causal structure of spacetime and Scott topology

In this paper we establish the spacetime manifold as a partially ordered set via the casual structure. We show that these partially ordered sets are naturally continuous as a suitable way below relation can be established via the chronological order. We further consider those classes of spacetimes on which a lattice structure can be endowed by physically defining the joins and meets. By considering the physical properties of null geodesics on the spacetime manifold we show that these lattices are necessarily distributive. These lattices are then continuous as a result of the equivalence between the way below relation and chronology. This enables us to define the Scott topology on the spacetime manifold and describe it on an equal footing as any other continuous lattice. We further show that the Scott topology is a proper subset of Alexandroff topology, which must be the manifold topology for the strongly causal spacetimes, (and hence a coarser topology than Alexandroff). In the process we find some interesting results on the sobriety of these manifolds. We prove that they are necessarily not sober under the Scott topology but regain their sobriety under Alexandroff topology. We also define a dual Scott topology on these manifolds by endowing them with bicontinuous poset structure and show that the join of the Scott topology with the dual is the Alexandroff topology. We also discuss the previous works done in this topic and how the present work generalises those results to some extent.

The language of knowledge spaces in pre-topology1

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.

Topologies, posets and finite quandles

An Alexandroff space is a topological space in which every intersection of open sets is open. There is one to one correspondence between Alexandroff T0 -spaces and partially ordered sets (posets). We investigate Alexandroff T0 -topologies on finite quandles. We prove that there is a non-trivial topology on a finite quandle making right multiplications continuous functions if and only if the quandle has more than one orbit. Furthermore, we show that right continuous posets on quandles with n orbits are n-partite. We also find, for the even dihedral quandles, the number of all possible topologies making the right multiplications continuous. Some explicit computations for quandles of cardinality up to five are given.

On the triviality of flows in Alexandroff spaces

We prove that the unique possible flow in an Alexandroff T0-space is the trivial one. To motivate this result, we relate Alexandroff spaces to topological hyperspaces.

ODigraphic Topology On Directed Edges

In this paper, we study the odigraphic topology for a directed edges of a digraph. We give some properties of this topology, in particular we prove that is an Alexandroff topology and when two digraphs are isomorphic, their odigraphic topologies will be homeomorphic. We give some properties matching digraphs and homeomorphic topology spaces. Finally, we investigate the connectedness of this topology and some relations between the connectedness of the digraph and the topology .

A topological approach to rough sets from a granular computing perspective

There are mainly four kinds of rough set models: the binary relation model, the neighborhood operator/system model, the covering model and the subset system model. While in a broad sense, all of these models are related to topological structures. For this reason, this paper selects Alexandrov topology as the fundamental structure to construct approximation operators and to investigate rough sets from a granular computing perspective. By using the notions of open-hoods and closed-hoods, two zooming extension operators, two zooming intension operators, two zooming-in operators and four zooming-out operators are defined and the composition/decomposition problems are studied. These models can be considered as a common framework of the preordered relation model, the Alexandrov neighborhood operator model and the covering model of rough sets.

An intuitionistic set-theoretical model of fully dependent CC

AbstractWerner’s set-theoretical model is one of the simplest models of CIC. It combines a functional view of predicative universes with a collapsed view of the impredicative sort “ ${\tt Prop}$ ”. However, this model of ${\tt Prop}$ is so coarse that the principle of excluded middle $P \lor \neg P$ holds. Following our previous work, we interpret ${\tt Prop}$ into a topological space (a special case of Heyting algebra) to make the model more intuitionistic without sacrificing simplicity. We improve on that work by providing a full interpretation of dependent product types, using Alexandroff spaces. We also extend our approach to inductive types by adding support for ${\mathsf{list}}$ s.

ON A TOPOLOGY DEFINED BY PRIMITIVE WORDS

Let f:X⟶X be a mapping. Consider P(f)={O⊆X:f-1(O)⊆O}. Then P(f) is an Alexandroff topology. A topological space X is called a primal space if its topology coincides with a P(f) for some mapping f:X⟶X. Let A be an alphabet and A* be the set of all finite words over A. A word is called primitive if it is not empty and not a proper power of another word. Let u be a nonempty word; then there exists a unique primitive word z and a unique integer k≥1 such that u=zk; z is called the primitive root of u; we denote by z=pA(u). It is convenient to set pA(εA)=εA, where εA is the empty word over A. By a primitive primal space we mean a space X that is homeomorphic to the subspace A+ of A* equipped with the topology P(pA) for some alphabet A. Our main result provides a structure theorem of primitive primal spaces.

GRAPHIC TOPOLOGY ON FUZZY GRAPHS

In this paper, we study the graphic topology $\mathcal{T}_{G}$ for a fuzzy graph. We give some properties of this topology, in particular we prove that $\mathcal{T}_{G}$ is an Alexandroff topology and when two graphs are isomorphic, their graphic topologies will be homeomorphic. We give some properties matching graphs and homeomorphic topology spaces. Finally, we investigate the connectedness of this topology and some relations between the connectedness of the graph and the topology $\mathcal{T}_{G}$.

A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean $$\ell$$-Algebras

Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame $$\mathcal {L}_B$$ generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of $$\mathcal {L}_B$$ is a canonical extension of B. Our main result generalizes this approach to the category $$\varvec{ ba \ell }$$ of bounded archimedean $$\ell$$ -algebras, thus yielding a point-free construction of canonical extensions in $$\varvec{ ba \ell }$$ . We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean $$\ell$$ -ideals of A is a canonical extension of $$A\in \varvec{ ba \ell }$$ , thus providing a generalization of the result of Holliday to $$\varvec{ ba \ell }$$ .

Z-graphic topology on undirected graph

In this work, we define ZG a topology on the vertex set of a graph G which preserves the connectivity of the graph, called Z-graphic topology. We prove that two isomorphic graphs have homeomorphic and symmetric Z-graphic topologies. We show that ZG is an Alexandroff topology and we give a necessary and sufficient condition for a topology to be Z-graphic.