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Our aim is to obtain a suitable characterization of certain topological properties of a random closed set through its capacity functional. The main technique mixes two different fields: on the one hand, the abstract simplicial complex associated to a covering, whose topology coincides with the topology of the original set thanks to the nerve theorem. On the other hand, the celebrated Choquet-Kendall-Matheron theorem, which states that a random closed set is characterized by its capacity functional.

Let $\kappa$ be an uncountable cardinal with $\kappa=\kappa^{{\kappa^+$'' implies the statement "every closed subset of ${}^\kappa\kappa$ is a continuous image of ${}^\kappa(\kappa^+)$'' or its negation., Comment: 30 pages

The purpose of this paper is to define and study a new class of sets called micro generalized closed sets and define micro generalized continuous function and irresolute micro generalized continuous function in micro topological spaces. Basic properties of micro generalized closed sets and its characterizations are analyzed.

In this paper, the properties of ψgs-closed sets in bitopological spaces are investigated. The relationships between ψgs-closed set and other closed sets in bitopological spaces are established and some properties of ψgs-closure and ψgs-interior are provided.

In this paper, we generalize the concept of a monotone convergence space (also called a d-space) to the setting of a Q -cotopological space, where Q is a commutative and integral quantale. We establish a D-completion for every stratified Q -cotopological space, which is a category reflection of the category S Q -CTop of stratified Q -cotopological spaces onto the full subcategory S Q -DCTop of monotone convergence Q -cotopological spaces. By introducing the notion of a tapered set, a direct characterization of the completion is obtained: the D-completion of each stratified Q -cotopological space X consists exactly of those tapered closed sets in X. We show that the D-completion can be applied to obtain a universal fuzzy directed completion of a Q -ordered set by endowing it with the Scott cotopology, taking the D-completion, and then passing to the specialization Q -order. Consequently, the category Q -DOrd of fuzzy directed complete Q -ordered sets and Scott continuous functions is reflective in the category Q -Ordσ of Q -ordered sets and Scott continuous functions.

Nano JD closed sets, a new class of Nano generalized closed sets in Nano topological spaces, relies on the g-interior and g closure operators. This paper aims to interpret and investigate its extension and connection with other existing concepts.

We extend, to the framework of topological vector spaces, two results by Horvath and Kuratowski related to conditions for a family of closed sets to have compact and nonempty intersection. This extension enables us to introduce a number of applications such as the existence of maximal elements in preordered spaces, issues related to KKM functions, fixed point theorems, a variant of a matching theorem by Fan, and mainly the improvement of some minimax and variational inequalities.

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. We introduce commutatively closed graphs and investigate properties of commutatively closed subsets of a ring with the help of graph theory. A particular attention is paid to constructions of subsets of a given diameter. In particular, we compute the diameter of artinian semisimple algebras.

The aim of the present article is to introduce the concepts of $$s_{p}-$$ convergent sequence in generalized intuitionistic fuzzy metric spaces, i.e., $${\mathscr {L}}$$ -fuzzy metric spaces and analyze relations of convergence, $$s_{p}-$$ convergence, $$s_{\infty }-$$ convergence and $$st-$$ convergence in $${\mathscr {L}}$$ -fuzzy metric spaces. Further, we examine $$s_{p}-$$ convergence, $$s_{\infty }-$$ convergence and $$st-$$ convergence using the subsequence of convergent sequence for above-stated spaces. We finally define $$s_{p}-$$ closed sets, $$s_{\infty }-$$ closed sets and $$st-$$ closed sets in intuitionistic fuzzy metric spaces and investigate relations of them.

It is known that the Hurwitz zeta-function $$\zeta(s,\alpha)$$ with transcendental or rational parameter $$\alpha$$ has a discrete universality property; i.e., the shifts $$\zeta(s+ikh,\alpha)$$ , $$k\in\mathbb N_0$$ , $$h> 0$$ , approximate a wide class of analytic functions. The case of algebraic irrational $$\alpha$$ is a complicated open problem. In the paper, some progress in this problem is achieved. It is proved that there exists a nonempty closed set $$F_{\alpha,h}$$ of analytic functions such that the functions in $$F_{\alpha,h}$$ are approximated by the above shifts. Also, the case of certain compositions $$\Phi(\zeta(s,\alpha))$$ is discussed.

الفكرة الرئيسية لهذا العمل الحالي هي تقديم أنواع جديدة من المجموعات فوق المغلقة في فضاءات فوق طوبولوجية تسمى supra -closed (باختصار ، مغلقة). علاوة على ذلك ، قارن هذه الفئة من المجموعات بأنواع أخرى من المجموعات أعلاه في الفضاءات فوق الطوبولوجية. دراسة وإثبات بعض صفاتهم.

We study the problem of finding the nearest $$\varOmega $$ Ω -stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $$\varOmega $$ Ω . Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

We give a complete characterization of closed sets F ⊂ ℝ2 whose distance function dF:= dist(·, F) is DC (i.e., is the difference of two convex functions on ℝ2). Using this characterization, a number of properties of such sets is proved.

Infra soft topology is one of the recent generalizations of soft topology which is closed under finite intersection. Herein, we contribute to this structure by presenting two kinds of soft covering properties, namely, infra soft compact and infra soft Lindelöf spaces. We describe them using a family of infra soft closed sets and display their main properties. With the assistance of examples, we mention some classical topological properties that are invalid in the frame of infra soft topology and determine under which condition they are valid. We focus on studying the “transmission” of these concepts between infra soft topology and classical infra topology which helps us to discover the behaviours of these concepts in infra soft topology using their counterparts in classical infra topology and vice versa. Among the obtained results, these concepts are closed under infra soft homeomorphisms and finite product of soft spaces. Finally, we introduce the concept of fixed soft points and reveal main characterizations, especially those induced from infra soft compact spaces.

The concept of neutrosophic continuous function was very first introduced by A.A. Salama et al. The main aim of this paper is to introduce a new concept of Neutrosophic continuous function namely Strongly Neutrosophic gsα* - continuous functions, Perfectly Neutrosophic gsα* - continuous functions and Totally Neutrosophic gsα* - continuous functions in Neutrosophic topological spaces. These concepts are derived from strongly generalized neutrosophic continuous function and perfectly generalized neutrosophic continuous function. Several interesting properties and characterizations are derived and compared with already existing neutrosophic functions.

In this paper, firstly we introduce the notions of soft supra locally α -closed sets in soft supra topological spaces. We investigate the relationships with different types of subsets of soft supra...

The aim of the paper is to introduce and investigate the concepts of pentapartitioned neutrosophic generalized semi-closed sets and pentapartitioned neutrosophic generalized semi-open sets in pentapartitioned neutrosophic topological spaces.

Let F be a smooth Riemann surface foliation on M∖E, where M is a complex manifold and E⊂M is a closed set. Fix a hermitian metric g on M∖E and assume that all leaves of F are hyperbolic. For each l...

In this article, we will define the new notions (e.g., b − θ -neighborhood system of point, b − θ -closure (interior) of a set, and b − θ -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying b − θ -closed set is fuzzifying γ -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying b − θ -derived set and fuzzifying b − θ -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly b − θ -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly b − θ -continuous mapping, fuzzifying strongly b − θ -irresolute mapping, and fuzzifying weakly b − θ -irresolute mapping along with different conditions for their existence are obtained.

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

In this article, the notions of partially continuous mappings, partially bounded, partially compact and partially closed sets of an ordered E-metric space are defined. These notions are used to introduce partial E-measure of noncompactness and prove the fixed point results of monotone mappings and sum of two monotone mappings. In this way we generalize many results including the well known results of Schauder, Darbo and Krasnoselskii, in the settings of ordered E-metric and ordered Banach spaces. We also provide nontrivial examples and existence results for a class of integral equations to validate the significance of our theory and results.

Florentin eSmarandache egeneralized ethe eintuitionistic efuzzy esets eto eNeutrosophic eset etheory ein e1998 eas ea enew ebranch eof ephilosophy. eA.A. eSalama eintroduced ethe econcept eof eNeutrosophic etopological espaces eby eusing ethe eNeutrosophic ecrisp esets. eIn ethis epaper, ewe eintroduce eand estudy ea enew eclass eof eNeutrosophic egeneralized eset, enamely eNeutrsophic epre egeneralized eregular eα- eclosed eset ein eNeutrosophic etopological espaces. eSome einteresting epropositions ebased eon ethis eset eare eintroduced eand established ewith esuitable examples eand etheir eproperties eare ealso ediscussed.

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domain Ω ⊂ ℝn, n ≥ 2. It is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain (for n ≥ 2). For 3D model, it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.

In this paper, we do further investigations on the statistical inner and outer limits of sequences of closed sets in metric spaces, which were introduced by Nuray, Rhoades, and Talo, Sever, Başar, and generalize the conventional Painleve-Kuratowski inner and outer limits. Also, we provide criteria for checking statistical Wijsman and Hausdorff set convergences and we examine the relationship between Kuratowski and Wijsman statistical convergence. A closer look on the concept of statistical Cauchyness, with respect to the Hausdorff “extended” metric h, completes this research.

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point of each modulus can be attained as the maximum modulus set of a quasiregular map. These examples are all of polynomial type. We also show that, subject to an additional constraint, such sets can even be attained by quasiregular maps of transcendental type.

We investigate when a mean field-type control system can fulfill a given constraint. Namely, given a closed set of probability measures on the torus, starting from any initial probability measure belonging to this set, does there exist a solution to the mean field control system remaining in it for any time? This property—the so-called viability property—is equivalently characterized through a property involving normals to the given set of probability measures. We prove that, if the Hamiltonian is nonpositive at any normal distribution to the given set, then the feedback strategy realizing the extremal shift rule provides the approximate viability. This implies the usual viability property. Conversely, the Hamiltonian is nonpositive at any normal distribution if the given set is viable. Our approach enables us to derive generalized feedback laws which ensure the trajectory to fulfill the constraint. This generalized feedback called here extremely shift rule is inspired by constructive motions developed by Krasovskii and Subbotin for differential games.

The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed sets, closure, and interior - they just so happen to coincide. The agreement between these four structures arising from a metric function d : X × X → [ 0 , ∞ ] is due to a combination of the metric axioms and the lattice structure of [ 0 , ∞ ] . Further motivation materializes from Lawvere's observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in [ 0 , ∞ ] . Metric spaces taking values in structures other than [ 0 , ∞ ] are relevant for generalizations of metric spaces and find a natural home in Lawvere's categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between [ 0 , ∞ ] and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in Q one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q-valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q-valued spaces as we investigate these issues.

We introduce some variants of contra-continuity in terms of ΛI -closed sets, namely contra-ΛI - continuous, contra quasi-ΛI -continuous and contra ΛI -irresolute functions. The relationships between these functions are investigated and their respective characterizations are established. Moreover, we study their behavior of several topological notions under the direct and inverse images of these functions.

We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Our algorithm is implemented in the language of the computer algebra system GAP 4. We discuss the implementation and give runtimes on some sample inputs. The classification of the closed subsets of an irreducible root system is closely related to the classification of the regular subalgebras, up to inner automorphism, of the corresponding simple Lie algebra. We show how to obtain the regular subalgebras corresponding to a given closed set for the root systems of rank 3. The version of this paper on the arxiv has tables describing the subalgebras that were obtained.