Topological Space

Neutrosophic Ranked Soft Sets (NRSS) provide a robust mathematical framework for managing uncertainty, imprecision, and parameter prioritization in complex decision environments. This paper presents a comprehensive study on the application of NRSS in intelligent computing and data-driven decision-making systems. By integrating neutrosophic logic with ranked soft set theory, the proposed model effectively captures the degrees of truth, uncertainty, and falsity along with their relative weight of parameters. We establish the theoretical foundations of NRSS and illustrate its utility through real-world-inspired scenarios in areas such as expert systems, intelligent filtering, and multi-criteria evaluation. The findings demonstrate that NRSS significantly enhance the adaptability, interpretability, and reliability of intelligent systems, making them well-suited for modern decision support applications.

A function ψ : [X] 2 → X is a called a weak selection if ψ({x, y}) ∈ {x, y} for every x, y ∈ X. To each weak selection ψ, one associates a topology τ ψ , generated by the sets (←, x) = {y = x: ψ(x, y) = y} and (x, →) = {y = x: ψ(x, y) = x}. Answering a question of S. García-Ferreira and A.H. Tomita [S. García-Ferreira, A.H. Tomita, A non-normal topology generated by a two-point selection, Topology Appl. 155 (10) (2008) 1105-1110], we show that (X, τ ψ ) is completely regular for every weak selection ψ. We further investigate to what extent the existence of a continuous weak selection on a topological space determines the topology of X. In particular, we answer two questions of V. Gutev and T. Nogura [V. Gutev, T. Nogura, Selection problems for hyperspaces, in: E. Pearl (Ed.), Open Problems in Topology 2, Elsevier B. V., 2007, pp. 161-170].

Social situations can be modeled as games with payoff struc- tures. The complexity of the situation emerges from the pattern of pay- offs. Inferring the outcome of the game from the payoff matrix is the conventional approach to describing, predicting or prescribing behaviour of participants. This approach makes a strong assumption of rationality of the players and suffers from other drawbacks as well. While Nash (8) proved a mixed equilibrium always exists, this appealing universality comes at a price. Mixed equilibria are not a natural and credible model of decision-making behaviour. Indeed, it has been shown that efficient algorithms to compute mixed equilibria likely do not exist for all games. (3) The pure Nash equilibrium, a more appealing model of behaviour, fails on universality: a subset of games - and a growing proportion as the number of players and choices increases - do not possess a pure Nash equilibrium. Miller (5),(6) describes unpublished research in which he took a ...

Real physical space, as a purely mathematical structure formed according to the rules of set theory, topology, and fractal geometry, was proposed by Michel Bounias (1943-2003) and the author. It emerges as a mathematical lattice of primary topological balls, which was named a tessellattice, and the size of a cell/ball in the tessellattice is comparative with the Planck length, ~10^{-35} m. Discrete fractal properties of the tessellattice allow the prediction of scales at which submicroscopic to cosmic structures should occur. This approach allows the development of a submicroscopic concept of physics, which describes Nature at a much deeper level than offered by the quantum-mechanical formalism developed at the atom scale, ~10^{-10} m. In addition, the approach makes it possible to define such fundamental physical notions as mass and charge from first submicroscopic principles, and this actually means that fundamental mathematics lays down the basic concepts of physics.

A simple way of obtaining separable quotients in the class of weakly countably determined (WCD) Banach spaces is presented (Theorem 1). A large class of Banach lattices, possessing as a quotient c 0 , l 1 , l 2 , or a reflexive Banach space with an unconditional Schauder basis, is indicated (Theorem 2).

Let X n,k be a Khalimsky topological n dimensional subspace with digital k-connectivity. In relation to the classification of spaces X n,k , by comparing several kinds of continuities and homeomorphisms, the paper proposes a category which is suitable for studying the spaces X n,k .

The existence of minimal and maximal fixed points for monotone operators defined on probabilistic Banach spaces is proved. We obtained sufficient conditions for the existence of coupled fixed point for mixed monotone condensing multivalued operators.

The aim of this paper is to present a general method for constructing natural tessellations of conceptual spaces that is based on their topological structure. This method works for a class of spaces that was defined some 80 years ago by the Russian mathematician Pavel Alexandroff. Alexandroff spaces, as they are called today, are distinguished from other topological spaces by the fact that they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, Rumfitt used a special case of Alexandroff's method to elucidate the logic of vague concepts in a new way. Elaborating his approach, it can be shown that the color spectrum -and similarly other conceptual spaces -give rise to classical systems of concepts that have the structure of atomic Boolean algebras. Rumfitt is not the first of having investigated conceptual systems of conceptual spaces. Since some time, due to Gärdenfors and other authors, conceptual spaces have become a popular modeling tool in cognitive psychology, artificial intelligence, linguistics, and philosophy. The core idea of the conceptual space approach is that concepts can be represented geometrically as regions in similarity spaces. Using prototypes and the underlying metric of similarity spaces Gärdenfors constructs a geometrical discretization of conceptual spaces by so-called Voronoi tessellations. As will be shown in this paper, his account can be interpreted as a geometrical version of Alexandroff's topological construction. More precisely, a Voronoi tessellation à la Gärdenfors is extensionally equivalent to a topological discretization constructed by Alexandroff's method. Even more, Rumfitt's as well as Gärdenfors's constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This suggests that the class of these Alexandroff spaces may provide a convenient framework for conceptual spaces in general.

The present paper deals with two types of topologies on the set of integers, Z: a quasi-discrete topology and a topology satisfying the T1 2 -separation axiom. Furthermore, for each n ∈ N, we develop countably many topologies on Z n which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on Z n , many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on.

In this paper, we introduce and study the notions of θ-g-derived, θ-g-border, θ-g-frontier and θ-g-exterior of a set via the notion of θ-g-open sets. Nakaoka and Oda ([9] and [10]) introduced the notion of maximal open sets and minimal closed sets. By the same token, we introduce new classes of sets called maximal θ-g-open sets, minimal θ-g-closed sets, θ-g-semi maximal open sets and θ-g-semi minimal closed sets and investigate some of their fundamental properties.

In this paper, we deal with the new class of pre-regular p-open sets in which the notion of preopen set is involved. We characterize these sets and study some of their fundamental properties. We also present two other notions called extremally p-discreteness and locally pindiscreteness by utilizing the notions of preopen and preclosed sets by which we obtain some equivalence relations for pre-regular p-open sets. Moreover, we define the notion of regular popen sets by utilizing the notion of pre-regular p-open sets. We investigate some of the main properties of these sets and study their relations to pre-regular p-open sets.

In this paper, we introduce and study some basic properties of new classes of sets called ψ-I -closed sets and weakly ψ-I -closed sets. Moreover, we offer a new type of contra-continuous function in the context of ideal topology called contra-ψ-I continuous function and present its fundamental properties.

Let n > 1, ω = 1(−1+i√3) and En = {a+bω|a,b ∈ Zn}. In this paper we find the order of the group of units of the ring En. We also charectrize von Neumann elements in En. In addition, we characterize the values of n for which En is a von Neumann regular or a von Neu- mann local ring. This paper is motivated by applications to computer arithmetic using finite factor rings, and public key cryptography based on groups of units and von Neumann elements of finite rings.

We provide a machinery for transferring some properties of metrizable AN R-spaces to metrizable LC n -spaces. As a result, we show that for complete metrizable spaces the properties ALC n , LC n and W LC n coincide to each other. We also provide the following spectral characterizations of ALC n and cell-like compacta: A compactum X is ALC n if and only if X is the limit space of a σ-complete inverse system S = {X α , p β α , α < β < τ } consisting of compact metrizable LC n -spaces X α such that all bonding projections p β α , as a well all limit projections p α , are U V n -maps. A compactum X is a cell-like (resp., U V n ) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., U V n ) metric compacta.

We discuss the question of extending homeomorphisms between closed subsets of the Cantor cube $D^{\tau }$ . It is established that any homeomorphism between two closed negligible subsets of $D^{\tau }$ can be extended to an autohomeomorphism of $D^{\tau }$ .

A characterization of n-dimensional spaces via continuous selections avoiding Z n -sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michael's result.

A selection theorem for set-valued maps into spaces with binary normal closed subbases is established. This theorem implies some results of A. Ivanov (see , ) concerning superextensions of k-metrizable compacta. A characterization of k-metrizable compacta in terms of usco retractions into superextensions and extension of functions is also provided.

We prove that a Hausdorff space X is very I-favorable if and only if X is the almost limit space of a σ-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very I-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces.

This paper explores the decomposition of pre-𝛽-irresolute (pre-𝛽-irr) maps and their connection to generalized closed sets (g-closed) in topological space (top. sp.). We provide the necessary background on top. sp., continuity, closed, and related concepts. We introduce the definitions of pre-𝛽-irr maps and g-closed and discuss their basic properties. The decomposition theorem is presented, stating that every pre-𝛽-irr map can be decomposed into a composition of a pre-𝛽-continuous (pre-𝛽-cont.) map and a b-irr map. Examples and applications are provided to illustrate the behavior of pre-𝛽-irr maps and the practical implications of the decomposition theorem. We examine the relationships between pre-𝛽-cont. maps and pre-𝛽-irr maps, exploring the conditions under which a pre-𝛽-irr map is also 𝑏-irr, and provide a proof for this relationship. The properties of pre-𝛽-cont. maps, 𝑏maps, g-pre-𝛽-cont. maps, and g-𝑏-cont. maps are discussed, along with proofs and examples to illustrate their applicability. This paper contributes to the understanding of pre-𝛽-irr maps, their decomposition, and their relationships with gclosed, providing a foundation for further research in top. sp. and map decompositions.

Cell structures were introduced by W. Debski and E. Tymchatyn as a way to study some classes of topological spaces and their continuous functions by means of discrete approximations. In this work we weaken the notion of cell structure and prove that the resulting class of topological space admitting such a generalized cell structure includes non-regular spaces.

Some of the properties of vector-valued fuzzy multifunctions are studied. The notion of sum fuzzy multifunction, convex hull fuzzy multifunction, close convex hull fuzzy multifunction, and upper demicontinuous are given, and some of the properties of these fuzzy multifunctions are investigated.

A variety of mathematical frameworks—such as fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, soft sets, rough sets, and plithogenic sets—have been developed to model uncertainty, with wide applications in decision making, data analysis, and artificial intelligence. Within soft set theory, extensions like hypersoft sets, indeterm-soft sets, indeterm-hypersoft sets, bipolar soft sets, and bipolar hypersoft sets have further enhanced its expressive power. In this paper, we introduce two new constructs: bipolar indeterm-soft sets and bipolar indeterm-hypersoft sets. We provide their formal definitions, establish key algebraic properties, and demonstrate how they naturally combine bipolar evaluation with inherent indeterminacy. These models offer a versatile toolkit for capturing complex forms of uncertainty and lay the groundwork for future theoretical advances and practical applications in soft set theory.

We investigate ternary products of groupoids and prove that there is a one-to-one correspondence between the collection of right modular groupoids with a left identity element l and laterally commutative, l-bi-unital semiheaps. This result is applied to prove that the natural ternary product induced by a right modular groupoid S with left identity is isomorphic to the natural ternary product of a groupoid T if and only if S and T are isomorphic groupoids. Ternary products of other classes of groupoids are also characterised, including the natural and standard ternary products of inverse semigroups. These two classes of ternary products are proved to be varieties of type (3,1,1).

We prove that the set of all orders of finite algebras in the groupoid variety of anti-rectangular Abel Grassmann bands consists of all powers of four. We also prove that any groupoid anti-isomorphic to a finite or countable anti-rectangular Abel Grassmann band G is isomorphic to G. It is proved that within isomorphism there is only one countable anti-rectangular Abel Grassmann band and that it is isomophic to a proper subgroupoid of itself.

We investigate ternary products of groupoids and prove that there is a one-to-one correspondence between the collection of right modular groupoids with a left identity element l and laterally commutative, l-bi-unital semiheaps. This result is applied to prove that the natural ternary product induced by a right modular groupoid S with left identity is isomorphic to the natural ternary product of a groupoid T if and only if S and T are isomorphic groupoids. Ternary products of other classes of groupoids are also characterised, including the natural and standard ternary products of inverse semigroups. These two classes of ternary products are proved to be varieties of type (3,1,1).

Volume 83 of “Neutrosophic Sets and Systems” features a diverse collection of research applying neutrosophic theory to address complex challenges across various fields, including information science, engineering, and healthcare. The papers in this volume present new models and methodologies that effectively handle uncertainty, indeterminacy, and imprecision. Significant theoretical contributions include the exploration of concepts like Neutrosophic Metric Spaces, Fermatean Neutrosophic Graphs, and SuperHyperSoft Sets, along with their applications in fields like energy supply systems and fixed-point theorems. The volume also highlights practical applications in multi-criteria decision-making (MCDM) for areas such as entrepreneurial education and civil litigation efficiency. Furthermore, research is presented on the use of neutrosophic logic with advanced technologies like deep learning and machine learning for tasks such as brain tumor medical image analysis and sleep disorder prediction. The collection also includes studies on risk management, digital media arts evaluation, and natural language processing. This volume demonstrates the expansive and evolving utility of neutrosophic theory in both foundational research and real-world problem-solving.

Volume 81 of “Neutrosophic Sets and Systems” features diverse applications of neutrosophic theory across various domains of science and engineering. The collection of papers explores advanced methodologies and models for addressing complex problems characterized by uncertainty, imprecision, and vagueness. A primary focus is on the development and application of neutrosophic frameworks for multi-criteria decision-making (MCDM), with studies evaluating teaching quality in higher education, assessing service quality in tourism, and analyzing the performance of industrial systems such as the new energy vehicle supply chain. The volume also includes theoretical contributions, such as research on neutrosophic graphs and their connectivity, the use of different types of neutrosophic sets like Type-2, Triangular, and HyperSoft Sets, and a bibliometric analysis of the journal itself. These papers demonstrate the versatility of neutrosophic theory as a tool for solving real-world challenges, including the evaluation of landscape design for abandoned coal mine sites, risk assessment of municipal projects, and the prioritization of higher education management strategies for sustainability.

“Neutrosophic Sets and Systems” (Volume 68) presents a comprehensive collection of peer reviewed contributions that advance the theory and application of neutrosophy, neutrosophic sets, logic, probability, and statistics. Building on the foundational concept that every proposition can be simultaneously true, false, and indeterminate to varying degrees, the issue showcases novel mathematical frameworks—such as rough statistical convergence in neutrosophic normed spaces, q rung neutrosophic topologies, and neutrosophic metric structures—and demonstrates their utility across diverse domains including decision making, image segmentation, data mining, and artificial intelligence. The volume also highlights interdisciplinary extensions, ranging from fuzzy type extensions and intuitionistic variants to applications in medical AI, database decomposition, and multi criteria evaluation. Emphasizing rigorous proof techniques, illustrative examples, and computational perspectives, the issue underscores the growing relevance of neutrosophic methodologies for handling incomplete, inconsistent, and indeterminate information in modern scientific and engineering problems.

This volume of “Neutrosophic Sets and Systems”, Volume 66, presents a collection of articles that advance both the theoretical foundations and practical applications of neutrosophic theory. The research explores the generalization of neutrosophic concepts to new structures, such as neutrosophic metric spaces and neutrosophic INK-algebras. It also demonstrates the versatility of neutrosophic sets in addressing complex, real-world problems. Key contributions include the application of neutrosophic cognitive maps for clinical decision-making, the integration of neutrosophic theory with bioinformatics algorithms, and the development of intelligent neutrosophic models for tasks like selecting IoT service providers. This volume underscores the role of neutrosophy in providing a framework to handle uncertainty and indeterminacy in a wide range of scientific and engineering disciplines.

Volume 61 of “Neutrosophic Sets and Systems” presents a diverse collection of research focused on advancing the theoretical foundations and expanding the practical applications of neutrosophic theory across various domains. The papers explore new algebraic structures, including the properties of neutrosophic rings and matrices. They also introduce novel concepts in neutrosophic topology, such as neutrosophic n-normed linear spaces and weaker forms of bipolar neutrosophic open sets. The volume highlights a wide range of applications, including the use of neutrosophic sets in medicine and healthcare, e-commerce site evaluation, and business analytics. Other contributions include a new method for classroom teaching evaluation using machine vision and neutrosophic sets, and the application of neutrosophic systems to solve problems in text analysis, drug diffusion, and optimization. The research collectively demonstrates the utility of neutrosophic structures in handling real-world problems characterized by uncertainty, incompleteness, and indeterminacy.

Volume 60 of “Neutrosophic Sets and Systems” is an international journal that publishes advanced studies and applications of neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, and neutrosophic statistics in various fields. The papers included in this volume develop new theories and explore applications of these neutrosophic concepts in areas such as algebra, geometry, and topology. The journal is a generalization of fuzzy sets and logic, and its defining characteristic is the study of "neutralities," which are ideas that are neither true nor false. It is indexed in several databases, including SCOPUS and Google Scholar.

Volume 54 of “Neutrosophic Sets and Systems” presents a comprehensive collection of original research devoted to the rapidly expanding field of neutrosophy and its many mathematical extensions. The special issue focuses on neutrosophic algebraic structures, refined neutrosophic systems, and intuitionistic neutrosophic crisp topologies, showcasing contributions primarily from scholars in the Arab world. Articles develop new theoretical frameworks—such as 3 refined neutrosophic real functions, inner product spaces, and number theoretic concepts—and introduce novel separation axioms for both supra  and infra topological spaces built on intuitionistic neutrosophic crisp sets. Together, these works advance the unification of classical algebra, topology, and logic under the neutrosophic paradigm, highlighting the versatility of neutrosophic sets in modeling indeterminacy, neutrality, and contradiction across diverse mathematical domains.

Volume 53 of “Neutrosophic Sets and Systems” (2023) gathers a diverse range of studies advancing both the theoretical foundations and applied dimensions of neutrosophy. The issue presents new developments in algebraic structures, topology, probability, statistics, and graph theory under neutrosophic frameworks, as well as innovative models such as hypersoft sets, cubic sets, and type-2 neutrosophic systems. Applications span multiple domains, including decision-making, medical diagnosis, urban traffic management, vehicular malfunction detection, queuing systems, supply chain inventory modeling, and image processing. Several contributions introduce computational algorithms, aggregation operators, and optimization methods, highlighting the growing integration of neutrosophy with engineering, computer science, and applied mathematics. Collectively, this volume underscores the flexibility of neutrosophic approaches in addressing uncertainty, indeterminacy, and inconsistency across interdisciplinary contexts.

“Neutrosophic Sets and Systems” is a peer reviewed, open access journal dedicated to advancing the theory and applications of neutrosophy, neutrosophic sets, neutrosophic logic, neutrosophic probability, and neutrosophic statistics. Since its inception in 1995, the journal has provided a multidisciplinary platform for researchers to explore extensions of fuzzy and intuitionistic fuzzy frameworks, introduce novel neutrosophic models, and apply these concepts across diverse domains such as engineering, medicine, computer science, economics, and social sciences. The Volume 51 continues this mission by publishing original research articles, methodological innovations, and comprehensive surveys. Highlights include: Development of single valued neutrosophic Kruskal Wallis and Mann Whitney statistical tests, illustrated through medical and educational case studies; Extensions of neutrosophic entropy measures, decision making algorithms, and machine learning classifiers that exploit neutrosophic scoring; Theoretical contributions on neutrosophic algebraic structures, refined neutrosophic number theory, and interval neutrosophic spaces. All articles are released under a Creative Commons Attribution License, ensuring unrestricted reuse with proper citation. The journal is indexed in Scopus, ESCI (Web of Science), DOAJ, Google Scholar, and numerous international bibliographic databases, reflecting its growing impact (2022 Impact Factor ≈ 1.74). By fostering rigorous, interdisciplinary research, “Neutrosophic Sets and Systems” aims to expand the frontiers of uncertainty modeling and provide robust tools for real world decision making.

“Neutrosophic Sets and Systems” (NSS) is a peer reviewed international journal dedicated to advanced research on neutrosophy, neutrosophic sets, logic, probability, statistics, and their applications across mathematics, engineering, computer science, medicine, and other domains. Established in 1995, the journal publishes original theoretical developments, methodological innovations, and practical case studies that extend fuzzy and intuitionistic fuzzy frameworks by incorporating degrees of truth, indeterminacy, and falsity. Volume 50 (2022) showcases contributions such as novel aggregation operators for simplified neutrosophic sets, multi criteria decision making techniques, and interdisciplinary applications ranging from teaching quality assessment to risk analysis. All articles are open access under a Creative Commons Attribution License, indexed in Scopus, ESCI, DOAJ, and numerous scholarly databases, reflecting the journal’s growing impact (2022 Impact Factor ≈ 1.74). The issue underscores the versatility of neutrosophic methodologies for modeling uncertainty, inconsistency, and partial truth in complex real world problems.

Volume 49 of “Neutrosophic Sets and Systems” (2022) presents a comprehensive collection of recent research developments in neutrosophic theory and its wide-ranging applications. The issue covers fundamental advances in neutrosophic sets, logic, probability, and statistics, as well as hybrid extensions such as interval-valued, bipolar, refined, pentapartitioned, and rough neutrosophic models. Contributions explore both theoretical foundations—through new algebraic structures, axioms, and entropy measures—and practical implementations across diverse fields, including medical diagnosis, decision-making, inventory management, graph theory, topology, engineering, environmental risk assessment, and Industry 4.0. By integrating uncertainty, indeterminacy, and inconsistency into formal reasoning systems, the articles collectively highlight the versatility of neutrosophic approaches in solving real-world problems where classical and fuzzy models fall short. This volume thus serves as a valuable reference for scholars and practitioners interested in the advancement of neutrosophic science and its interdisciplinary impact.

This volume of “Neutrosophic Sets and Systems” presents a collection of papers focused on advanced studies in neutrosophy, neutrosophic sets, neutrosophic logic, neutrosophic probability, and neutrosophic statistics. Neutrosophy is described as a new branch of philosophy that studies the origin, nature, and scope of neutralities, considering every notion with its opposite and a spectrum of neutralities in between. The document highlights that neutrosophic set and neutrosophic logic are generalizations of fuzzy set and fuzzy logic, respectively. The papers within this volume cover a wide range of applications, including a new business model for transportation, anti-neutrosophic multi-fuzzy ideals of near rings, cloud computing risk assessment, and analysis of heavy metal contamination in river water. It also discusses topics such as single-valued pentapartitioned neutrosophic dice similarity measures and interval-valued intuitionistic hypersoft sets for multi-criteria decision-making.

In 1992 Alexandru Nica published a highly influential paper examining the C*-algebras generated by a semigroup he calls a quasi-lattice ordered group. Of particular interest to Nica were the C*-algebra generated by the Toeplitz representation of a quasi-lattice order, and a universally defined C*-algebra he calls C*(G, P). Due to the concreteness of the first C*-algebra and the universal properties of the second, Nica was interested in finding sufficient conditions to determine when these two algebras were isomorphic. He called quasi-lattice orders that satisfied this isomorphism condition amenable. Finding techniques that establish amenability is now a topic at the cutting edge of research in the study of C*-algebras of semigroups. In this dissertation, we follow the outline set out by Nica's paper to study the C*-algebras generated by weakly quasi-lattice ordered groups. Most of the proofs throughout this dissertation required a substantial amount of original work to fill in details omitted by Nica, as he often only offered a proof outline.

In this paper, we introduce and study the concept of soft semi-‫-ܫ‬open sets with respect to an ideal. Then using these sets, we also study soft semi-‫-ܫ‬continuous functions, soft semi-‫-ܫ‬open functions and obtained some characterizations and several properties concerning these functions.

In this paper, we introduce and study the concept of soft semi-‫-ܫ‬open sets with respect to an ideal. Then using these sets, we also study soft semi-‫-ܫ‬continuous functions, soft semi-‫-ܫ‬open functions and obtained some characterizations and several properties concerning these functions.

As a tool to investigate almost periodic functions of several variables which is important for the study of asymptotic behavior of differential equations, we give answer to a question raised by Basit in 1971, which is an extension of the classical Bohl-Bohr theorem (to almost periodic functions with two or more variables). We also extend a theorem due to Loomis (who obtained it for functions with values in a finite dimensional space) to functions with values in Banach space E, under conditions introduced by Kadets. As an application, we obtain a result on the almost periodicity of a double integral of an almost periodic function defined on the plane.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so-called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the socalled Local-to-Global Principle that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to noncompact manifolds.