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We formulate two open problems related to and, in a sense, suggested by the Reiterman–Tholen characterization of effective descent morphisms of topological spaces.

It is shown that for any ordinal α, there exists a T 0-space whose d-rank is equal to α.

We obtain explicitly principal extensions of analytic functions of the complex variable into an infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We consider an extension of differentiable in the sense of Gâteaux functions with values in a topological vector space being an expansion of the mentioned algebra and its relations to spatial potentials.

Recently, J.~T.~Denniston, A.~Melton, and S.~E.~Rodabaugh introduced a lattice-valued analogue of the concept of institution of J.~A.~Goguen and R.~M.~Burstall, comparing it, moreover, with the (lattice-valued version of the) notion of topological system of S.~Vickers. In this paper, we show that a suitable generalization of topological systems provides a convenient framework for doing certain kinds of (lattice-valued) institutions.

We investigate which symmetric topological spaces have precisely one compatible nearness structure. We show how this situation exists for T1 topological spaces and for T2 topological spaces. The compatible nearness structures on a symmetric topological space form a sort of “interval” of structures with the topology itself as the topmost.

In this paper, our focus is to investigate the notion of irresolute topological vector spaces. Irresolute topological vector spaces are defined by using semi open sets and irresolute mappings. The notion of irresolute topological vector spaces is analog to the notion of topological vector spaces, but mathematically it behaves differently. An example is given to show that an irresolute topological vector space is not a topological vector space. It is proved that: 1) Irresolute topological vector spaces possess open hereditary property; 2) A homomorphism of irresolute topological vector spaces is irresolute if and only if it is irresolute at identity element; 3) In irresolute topological vector spaces, the scalar multiple of semi compact set is semi compact; 4) In irresolute topological vector spaces, every semi open set is translationally invariant.

Summary Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.

In this paper, an estimation of lower bound of topological entropy for coupled-expanding systems associated with transition matrices in compact Hausdorff spaces is given. Estimations of upper and lower bounds of topological entropy for systems in compact metric spaces are obtained by their topological equi-semiconjugacy to subshifts of finite type under certain conditions., 9 pages

This paper shows that given a certain frame L, the construct of strict L-bornological spaces, introduced by Abel and Sostak, is a topological universe.

We study the properties of quasihomeomorphisms and meet-semilattice equivalences of generalized topological spaces. Since the results of lattice equivalence of topological spaces were stated by the concept of closedness, so we give a generalization of those results for generalized topological spaces by defining closed sets by closure operators.

The classical Bourbaki–Alaoglu theorem asserts that the polar U ∘ of a neighborhood of 0 in a locally convex topological vector space V is a weak ⁎ compact subset of the dual space V ⁎ . G. Plotkin has proved an analogous theorem in the framework of continuous d-cones, a kind of asymmetric version of topological vector spaces in Domain Theory. In this paper we extend Plotkin's results. We consider topological spaces X with a finitary continuous algebraic structure in the sense of universal algebra instead of a cone structure. Linear functionals are replaced by continuous algebra homomorphisms into a test algebra R replacing the reals. Our main result, Theorem 5.8 , concerns the compactness of the space C ⁎ of continuous homomorphisms from X to R under appropriate hypotheses. We exhibit conditions under which C ⁎ is not only a space but also an algebra, as in the classical situation. This leads us to the notion of entropicity in the sense of universal algebra. The background for our investigation is Domain Theory and its use in denotational semantics (see [8] ). Thus our spaces are strongly non-Hausdorff. This paper can be seen as a contribution to asymmetric topology and analysis.

The aim of this paper is to introduce the concept of generalized topological molecular lattices briefly GTMLs as a generalization of Wang’s topological molecular lattices TMLs, Csaszar’s setpoint generalized topological spaces and lattice valued generalized topological spaces. Some notions such as continuous GOHs, convergence theory and separation axioms are introduced. Moreover, the relations among them are investigated.

The purpose of this paper is to introduce the concept of a (m;n) regular generalized star b-closed set in a bigeneralized topological space. Moreover, some basic properties and its relationship with the already existing class of closed sets are also studied.

This paper provides variable-basis lattice-valued analogues of the well-known results that the construct Prost of preordered sets, firstly, is concretely isomorphic to a full concretely coreflective subcategory of the category Top of topological spaces (which employs the concept of the dual of the specialization preorder), and, secondly, is (non-concretely) isomorphic to a full coreflective subcategory of the category TopSys of topological systems of S. Vickers (which employs the spatialization procedure for topological systems). Dualizing these results, one arrives at the similar properties of quasi-pseudo-metric spaces built over locales.

[EN] In the paper one defines topological invariants of type degree for morphisms in the category Top₍₂₎ of topological pairs of spaces and continuous single valued maps, which admit homological n-spheres as target and arbitrary topological pairs of spaces as source. The different described degrees are acquired by means homological methods, and are a powerful tool in the root theory. Several existence theorems are obtained for equations with multivalued transformations., Project supported by CNEPRU No B00920110092 and PNR / ATRST No 8/u250/713. Laboratory M.M.E.R.E. U.M.1 Constantine and Russian FBR grant 11-01 -00382-a.

In this chapter we present basic concepts and examples connected with topological vector spaces.

This paper describes a generalization of the Papert–Papert–Isbell adjunction between topological spaces and frames to a generalized one between the category of generalized topological spaces and the category of complete join-semilattices. Within these categories we aim to study certain separation axioms in a categorical point of view.

© 2014, The Author(s).In this paper, we continue to investigate some important results in generalized topological groups and we prove extension closed property for connectedness, compactness, and separability of generalized topological groups. Last, we define generalized topological group actions on generalized topological spaces and we establish a homeomorphism between action group and action space. [Figure not available: see fulltext.]

In this paper, we characterize monotonically countably paracompact (or monotonically countably metacompact) spaces by semi-continuous maps with values into some ordered topological vector spaces or some topological vector lattices. These extend earlier results for real-valued semi-continuous functions.

This paper shows that (L,M)-fuzzy topology of U. Hohle, T. Kubiak and A. Sostak is an instance of a general fuzzification procedure for topological categories, which amounts to the construction of a new topological category from a given one. This fuzzification procedure motivates a partial dualization of the machinery of tower extension of topological constructs of D. Zhang, thereby providing the procedure of tower extension of topological categories. With the help of this dualization, we arrive at the meta-mathematical result that the concept of (L,M)-fuzzy topology and the notion of approach space of R. Lowen are ''dual'' to each other.

In the present paper, we consider the problem of existence of nonequivalent definitions of topological transitivity, which is a classical problem of the topological dynamics. In particular, we use the fact that all available definitions of this kind imply a condition imposed on the dynamical system. The main result of our investigations is the complete classification of these dynamical systems.

In this paper, we define new classes of sets called preopen sets, θI-semi-open sets, θI-β-open sets and θI-α-open sets in ideal topological spaces. Also, by using these sets, we obtain new decompositions of continuity in ideal topological spaces.

For Hausdorff topological monoids, the concept of a unitary Cauchy net is a generalization of the concept of a fundamental sequence of reals. We consider properties and applications of such nets and of corresponding filters and prove, in particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commutative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the classical concepts of a completeness and of a completion into the theory of topological monoids.

In this paper we introduce topological and algebraic structures on the space of DNA sequences. The idea of these structures comes from genetic recombination.

In this paper, we introduce separation axioms in quad topological space (q topological spaces) and study some of their properties.

Based on the work of Ma and Wu,[9] some equivalent definitions of topological pressure of a semigroup of continuous maps are given and several of their basic properties are provided by using separated sets and spanning sets. We also answer an open problem of Biś and Urbanski.[19] That is, letting fi, i = 2, … , k, be homeomorphisms acting on a compact metric space, G1 = {idX, f2, … , fk}, G− 11 = {idX, f− 12, …, fk− 1} and letting G and G−1 denote the semigroups generated by G1 and G− 11, respectively, we give an example showing that the topological entropy of G does not equal the topological entropy of G−1.

In this paper we propose and explore a general notion of chaos in the abstract context of continuous actions of topological semigroups and show that any chaotic action on a Hausdorff uniform space is sensitive to initial conditions.

In this paper, we present a unified treatment on the linear, differentiable and topological equivalences for time-invariant linear control systems governed by ordinary differential equations (ODEs). Especially, we give a new solution to the topological classification problem for linear control systems by means of an elementary analytic approach, which differs considerably from J.C. Willems’ algebraic method developed in (Willems, 1980) [17] . More precisely, based on P. Brunovsky’s results in (Brunovsky, 1970) [2] , we determine all topological invariants for the linear control system. As an interesting by-product, we show that there exist only finitely many topological equivalence classes for any stabilizable system with given numbers of state and control variables, which might be useful for some engineering applications.

This paper looks at the properties of deficient topological measures, which are a generalization of topological measures. Integration of a real function that is continuous on a compact set with respect to a deficient topological measure is also investigated. The notions of r- and l-functionals are introduced and an analogue of the Riesz representation theorem is obtained for them. As corollaries, both well-known and new results for quasi-integrals and topological measures are presented (for example, a new version of the definition of a quasi-integral). Bibliography: 16 titles.

A new class of functions called strongly lower semi-continuous (SLSC) functions is defined and its properties are studied. It is shown that the arbitrary supremum and finite infimum of SLSC functions are again SLSC. Using these functions, an induced fuzzy topological space, called s-induced fuzzy topological space on a topological space (X, T), is introduced. Moreover, some incorrect results on fuzzy topological spaces obtained previously by some authors are identified and modified accordingly. Examples of the newly defined induced space are given and their various properties are investigated. Interrelationships between a fuzzy topological space (X, F) and the s-induced fuzzy topological space generated by the crisp members of F are examined. In this process, different lower semi-continuities and induced fuzzy spaces generated by them have been defined in a general set up and their few properties have been studied.

We introduce the different notions of a new class of continuous functions called generalized semi Lambda (gs) continuous function in topological spaces. Its properties and characterization are also discussed.

A twisted sum in the category of topological Abelian groups is a short exact sequence0→Y→X→Z→0where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to0→Y→Y×Z→Z→0. We study the classSTG𝕋of topological groupsGfor which every twisted sum0→𝕋→X→G→0splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups withℒ∞topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses ofSTG𝕋, we use the connection between extensions of the form0→𝕋→X→G→0and quasi-characters onG, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of𝒦-space, which were interpreted for topological groups by Cabello.

This paper is devoted to the existence of solutions for generalized variational like inequalities with fuzzy mappings in topological vector spaces by using a particular form of the generalized KKM-Theorem.

In this paper a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings begins. Some completeness and cocompleteness results are achieved. Generalized topological spaces help to reconstruct the important elements of the theory of locally definable and weakly definable spaces in the wide context of weakly topological structures., 26 pages