Alexandroff Spaces Research Articles

The Frame of Nuclei on an Alexandroff Space

Let $\mathcal {O} S$ be the frame of open sets of a topological space S, and let $N(\mathcal {O} S)$ be the frame of nuclei on $\mathcal {O} S$ . For an Alexandroff space S, we prove that $N(\mathcal {O} S)$ is spatial iff the infinite binary tree ${\mathscr{T}}_2$ does not embed isomorphically into (S,≤), where ≤ is the specialization preorder of S.

Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives

We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.

Rw*-closed sets in Alexandroff Spaces

This paper explained and defined the notion of regular weakly-star closed (briefly known as rw*-closed) sets in alexandroff spaces in which every point has a minimal neighbourhood. We discuss the characterizations and study their properties based on set theory along with the notion of rw*-open sets.

Completeness by Modal Definitions. Application to the Epistemic Logic With Hypotheses

We investigate the variant of epistemic logic S5 for reasoning about knowledge under hypotheses. The logic is equipped with a modal operator of necessity that can be parameterized with a hypothesis representing background assumptions. The modal operator can be described as relative necessity and the resulting logic turns out to be a variant of Chellas’ Conditional Logic. We present an axiomatization of the logic and its extension with the common knowledge operator and distributed knowledge operator. We show that the logics are decidable, complete w.r.t. Kripke as well as topological structures. The topological completeness results are obtained by utilizing the Alexandroff connection between preorders and Alexandroff spaces.

Alexandroff topologies and monoid actions

Abstract Given a monoid S acting (on the left) on a set X, all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X. Conversely, we prove that any Alexandroff topology may be obtained through a monoid action. Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions. Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets. Then, in relation to such a dependence, we study the family 𝒜 σ , X {\mathcal{A}_{\sigma,X}} of the closed subsets Y of X such that, for any y 1 , y 2 ∈ Y {y_{1},y_{2}\in Y} , there exists a third element y ∈ Y {y\in Y} whose closure contains both y 1 {y_{1}} and y 2 {y_{2}} . More in detail, relying on some specific properties of the maximal members of the family 𝒜 σ , X {\mathcal{A}_{\sigma,X}} , we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence. Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions.

The Relations between Residuated Frames and Residuated Connections

We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we introduce R-R (resp. D R - D R ) embedding maps and R-R (resp. D R - D R ) frame embedding maps.

Alexandroff Topology of Algebras Over an Integral Domain

Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R is lying over S and the localization of R with respect to $$S {\setminus } \{ 0 \}$$ is A. Let $${\mathbb {S}}$$ be the set of all S-nice subalgebras of A. We define a notion of open sets on $${\mathbb {S}}$$ which makes this set a $$T_0$$-Alexandroff space. This enables us to study the algebraic structure of $${\mathbb {S}}$$ from the point of view of topology. We prove that an irreducible subset of $${\mathbb {S}}$$ has a supremum with respect to the specialization order. We present equivalent conditions for an open set of $$\mathbb S$$ to be irreducible, and characterize the irreducible components of $${\mathbb {S}}$$. We also characterize quasi-compactness of subsets of a $$T_0$$-Alexandroff topological space.

Compactly Generated Spaces and Quasi-spaces in Topology

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $$\textsf {Top}$$ of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class $$\mathcal {C}$$ of objects we generalize the notion of $$\mathcal {C}$$-generated spaces, from which we derive, for instance, a general concept of Alexandroff spaces. Furthermore, as done for $$\textsf {Top}$$, we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.

On Completeness of Sliced Spaces under the Alexandrov Topology

We show that in a sliced spacetime ( V , g ) , global hyperbolicity in V is equivalent to T A -completeness of a slice, if and only if the product topology T P , on V, is equivalent to T A , where T A denotes the usual spacetime Alexandrov “interval” topology.

Various operations and right completeness in generalized residuated lattices

In this paper, based on generalized residuated lattices as an extension of Zhang’s complete residuated lattices, there are two types of structures: bi-partially orders, right (left) joins, right (left) complete lattices and right (left) Alexandrov topologies. We investigate their properties and the relationship between them. Moreover, monotone maps, right (left)-embedding maps and right-join (left-join) preserving maps are investigated with various operations as extensions of Zadeh powerset operations between these structures. As the foundation of fuzzy rough sets and fuzzy contexts, there exist adjunctions and Galois connections between maps from right(left) Alexandrov topologies to right(left) Alexandrov topologies. We give their examples.

Decomposition spaces and poset-stratified spaces

In 1920s R. L. Moore introduced upper semicontinuous and lower semicontinuous decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $\mathcal D$ of a topological space $X$ such that the decomposition spaces $X/\mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/\mathcal D$ is a poset, then the decomposition map $\pi:X \to X/\mathcal D$ is a continuous map from the topological space $X$ to the poset $X/\mathcal D$ with the associated Alexandroff topology, which is nowadays called a poset-stratified space. As an application, we capture the face poset of a real hyperplane arrangement $\mathcal A$ of $\mathbb R^n$ as the associated poset of the decomposition space $\mathbb R^n/\mathcal D(\mathcal A)$ of the decomposition $\mathcal D(\mathcal A)$ determined by the arrangement $\mathcal A$. We also show that for any locally small category $\mathcal C$ the set $hom_{\mathcal C}(X,Y)$ of morphisms from $X$ to $Y$ can be considered as a poset-stratified space, and that for any objects $S, T$ (where $S$ plays as a source object and $T$ as a target object) there are a covariant functor $\mathfrak{st}^S_*: \mathcal C \to \mathcal Strat$ and a contravariant functor $\mathfrak{st}^*_T$ $\mathfrak{st}^*_T: \mathcal C \to \mathcal Strat$ from $\mathcal C$ to the category $\mathcal Strat$ of poset-stratified spaces. We also make a remark about Yoneda's Lemmas as to poset-stratified space structures of $hom_{\mathcal C}(X,Y)$.

Primal topologies on the integers

Given an infinite set X and a function f : X → X, the primal topology on X induced by f is the topology τ f = {U ⊆ X : f −1(U ) ⊆ U}. In this paper, we prove that there are 2 ω pairwise non-homemomorphic primal topologies on ℕ. We also prove that an infinite set cannot have more than 2 ω pairwise non-homeomorphic primal topologies. We give a necessary and sufficient condition to guarantee that an Alexandroff topology be n-resolvable for an 2 ≤ n ∈ ℕ. Other results on primal topologies are also given.

A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies

Abstract In this paper, by means of the implication operator → on a completely distributive lattice M, we define the approximate degrees of M-fuzzifying convex structures, M-fuzzifying closure systems and M-fuzzifying Alexandrov topologies to interpret the approximate degrees to which a mapping is an M-fuzzifying convex structure, an M-fuzzifying closure system and an M-fuzzifying Alexandrov topology from a logical aspect. Moreover, we represent some properties of M-fuzzifying convex structures as well as its relations with M-fuzzifying closure systems and M-fuzzifying Alexandrov topologies by inequalities.

Interval topology in contact geometry

A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.

Interpolation and amalgamation in modal cylindric algebras

Let α be an ordinal and L be a unimodal logic (like S4 or S5). A modal cylindric algebra of dimension α, an LCA α, is a cylindric algebra of dimension α, expanded with α-many L modalities. For a frame (U, R) of L, each k < α, one defines a diamond box operator on . This defines the semantics of the L modalities in set algebras, with the rest of the operations defined like in cylindric set algebras of dimension α. We study interpolation properties for the corresponding predicate logic having α-many variables. Our results are valid for any reflexive L whose frames contain the universal frames (U, U × U ). In particular, they hold for K5CA α, S4CA α (which is an algebraizable extension of topological predicate logic with semantics induced by Alexandrov topologies).

Mθ - Closed Set In T0 -MAlexandroff Spaces

In this paper, we introduce definitions of Mθ-closed, Mθ-adherent and Mθ-open set, and we give some theorems and results related to these sets in minimal Alexandroff space.

The autohomeomorphism group of connected homogeneous functionally Alexandroff spaces

If is a map, then the family gives the closed sets of a topology P(f) on X called a functionally Alexandroff topology. If is a connected homogeneous functionally Alexandroff space, we give an algebraic characterization and some basic properties of its autohomeomorphism group .

Topologies induced by the representation of a betweenness relation as a family of order relations

Betweenness relations are the mathematical formalization of the geometrical notion of an element being in between other two elements. In this paper, we exploit a well-known result representing a betweenness relation as a family of order relations and analyse the corresponding family of induced (Alexandrov) topologies. In particular, the intersection of this family of topologies is proved to be the anti-discrete topology and a necessary and sufficient condition for the supremum of this family of topologies to be the discrete topology is provided. Interestingly, this condition is proved to hold when dealing with a finite set. We end with a discussion on the relation between the topology induced by an order relation or a metric and the family of topologies induced by the betweenness relation induced by the same order relation or metric.

Relations among (L, M)-fuzzy convex structures, (L, M)-fuzzy closure systems and (L, M)-fuzzy Alexandrov topologies in a degree sense

Relations among (L, M)-fuzzy convex structures, (L, M)-fuzzy closure systems and (L, M)-fuzzy Alexandrov topologies in a degree sense

Spectral primal spaces

Let [Formula: see text] be a mapping. Consider [Formula: see text] Then, according to Echi, [Formula: see text] is an Alexandroff topology. A topological space [Formula: see text] is called a primal space if its topology coincides with an [Formula: see text] for some mapping [Formula: see text]. We denote by [Formula: see text] the set of all fixed points of [Formula: see text], and [Formula: see text] the set of all periodic points of [Formula: see text]. The topology [Formula: see text] induces a preorder [Formula: see text] defined on [Formula: see text] by: [Formula: see text] if and only if [Formula: see text], for some integer [Formula: see text]. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function [Formula: see text] in order to get [Formula: see text] (respectively, the one-point compactification of [Formula: see text]) a spectral topology. More precisely, we show the following results. (1) [Formula: see text] is spectral if and only if [Formula: see text] is a finite set and every chain in the ordered set [Formula: see text] is finite. (2) The one-point(Alexandroff) compactification of [Formula: see text] is a spectral topology if and only if [Formula: see text] and every nonempty chain of [Formula: see text] has a least element. (3) The poset [Formula: see text] is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi–Naimi may be derived immediately from the general setting of the above results.