A duality of $κ$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $κ$-normability of a topological vector space implies its local convexity. There are given applications of $κ$-normed spaces for resolutions of differential equations and for approximations of functions in mathematical economy.
In this paper, we define precompact set in intuitionistic fuzzy metric spaces and prove that any subset of an intuitionistic fuzzy metric space is compact if and only if it is precompact and complete. Also we define topologically complete intuitionistic fuzzy metrizable spaces and prove that any $G_{δ}$ set in a complete intuitionistic fuzzy metric spaces is a topologically complete intuitionistic fuzzy metrizable space and vice versa. Finally, we define intuitionistic fuzzy normed spaces and fuzzy boundedness for linear operators and so we prove that every finite dimensional intuitionistic fu…
The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space $X$ is not a linear space, but merely a convex cone in the space of all linear functionals on $X.$ Due …
In this paper, we approach the question if some of the separation axioms are equivalent in the class of asymmetric normed spaces. In particular, we make a remark on a known theorem which states that every $T_1$ asymmetric normed space with compact closed unit ball must be finite-dimensional. We also explore the product structure of these spaces and characterize the topological (covering) dimension of all finite-dimensional asymmetric normed spaces.
Relying on Kolmogorov's classical characterization of normable Topological Vector spaces, we study the normability of those Probabilistic Normed Spaces that are also Topological Vector spaces and provide a characterization of normable Šerstnev spaces. We also study the normability of other two classes of Probabilistic Normed Spaces.
In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$…
This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space $(E, {\mathcal E}, μ)$ such…
We prove a reconstruction theorem for homeomorphism groups of open sets in metrizable locally convex topological vector spaces. We show that certain small subgroups of the full homeomorphism group obey the conditions of the above theorem.
We study modules over the ring $\widetilde{\C}$ of complex generalized numbers from a topological point of view, introducing the notions of $\widetilde{\C}$-linear topology and locally convex $\widetilde{\C}$-linear topology. In this context particular attention is given to completeness, continuity of $\widetilde{\C}$-linear maps and elements of duality theory for topological $\widetilde{\C}$-modules. As main examples we consider various Colombeau algebras of generalized functions
We study properties of strongly separately continuous mappings defined on subsets of products of topological spaces equipped with the topology of pointwise convergence. In particular, we give a necessary and sufficient condition for a strongly separately continuous mapping to be continuous on a product of an arbitrary family of topological spaces. Moreover, we charac\-terize the discontinuity point set of strongly separately continuous function defined on a subset of countable product of finite-dimensional normed spaces.
We present a fixed point theorem on topological cylinders in normed linear spaces for maps satisfying a property of stretching a space along paths. This result is a generalization of a similar theorem obtained by D. Papini and F. Zanolin. In view of the main result we discuss the existence of fixed points for maps defined on different types of domains and we propose alternative proofs for classical fixed point theorems, as Brouwer, Schauder and Krasnosel'ski\uı ones.
We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a se…
The motivation of this paper is a suggestion by Höle of comparing the notions of $\D$-boundedness and boundedness in Probabilistic Normed spaces (briefly PN spaces), with non necessarily continuous triangle functions. Such spaces are here called ``pre-PN spaces''. Some results on Šerstnev spaces due to B. Lafuerza, J. A. Rodriguez, and C. Sempi, are here extended to generalized Šerstnev spaces (these are pre-PN spaces satisfying a more general Šerstnev condition). We also prove some facts on PN spaces (with continuous triangle functions). First, a connection between fuzzy normed spaces defined…
We address two properties for Abelian topological groups: ``every closed subgroup is dually closed'' and ``every closed subgroup is dually embedded.'' We exhibit a pair of topological groups such that each has both of the properties and the product has neither, which refutes a remark of N. Noble. These examples are the additive group of integers topologized with respect to a convergent sequence as investigated by E.G. Zelenyuk and I.V. Protasov. The proof for the product relies on a theorem on exponential Diophantine equations.
In the context of categorical topology, more precisely that of T-categories [Hofmann, 2007], we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of "Stone duality", and show that Cauchy completeness of a T-category is precisely its sobriety.
For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify …
We show that the topology of uniform convergence on bounded sets is compatible with the group law of the automorphism group of a large class of spaces that are endowed with both a uniform structure and a bornology, thus yielding numerous examples of topological groups. This encompasses many of previously known cases, such as homeomorphism groups of locally compact spaces, automorphism groups of first-order structures, general linear groups of normed spaces, automorphism groups of uniform spaces, etc. We also study the automorphism group of a topological group within this framework (generalizin…
The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A special class of spacial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called cartesian and studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma \emph{I}, are called \emph{I}-cartesian and characterized. The characterization reveals a hidden structure of such sp…
In this paper we study the topological T-dual of spaces with a non-free circle action mainly using the stack theory method of Bunke and co-workers \cite{Bunke1}. We first compare three formalisms for obtaining the Topological T-dual of a semi-free $S^1$-space in a simple example. Then, we calculate the T-dual of general KK-monopole backgrounds using the stack theory method. We define the dyonic coordinate for these backgrounds. We introduce an approach to Topological T-duality using classifying spaces which simultaneously generalizes the methods of Bunke et al \cite{Bunke1} and Mathai and Wu \…
Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognizable subsets of the abstract algebra endowed with certain residuation operations. These results identify a connection between topological algebra as applied in algebra and Stone duality as applied in logic, and show that the notion of recognition originating in compute…
