Showing papers on "Topological space published in 2012"

Colored Tensor Models - a Review

Abstract: Colored tensor models have recently burst onto the scene as a promising con- ceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two- dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1=N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger{Dyson equations satisfying a Lie algebra (akin to the Virasoro al- gebra in two dimensions), non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.

Topological Vector Spaces and Distributions

Abstract: The traditional functional analysis deals mostly with Banach spaces and, in particular, Hilbert spaces. However, many classical vector spaces have canonical topologies that cannot be determined by a single norm. For example, many spaces of smooth functions, holomorphic functions, and distributions belong to the above class. Such spaces are the subject of the theory of topological vector spaces. Although the golden age of topological vector spaces was in the 1950ies, their theory is still evolving nowadays, contrary to a stereotyped view coming from incompetent sources. The current development of topological vector spaces is directed not so much towards general theory as towards applications in PDEs and in complex analytic geometry. We plan to discuss the basics of the theory of topological vector spaces, including some applications to distribution theory and to complex analytic geometry.

Some notes on soft topological spaces

Abstract: The first aim of this study is to define soft topological spaces and to define soft continuity of soft mappings. Second is to introduce soft product topology and study properties of soft projection mappings. Third is to define soft compactness and generalize Alexander subbase theorem and Tychonoff theorem to the soft topological spaces.

Neutrosophic Set and Neutrosophic Topological Spaces

Abstract: Neutrosophy has been introduced by Smarandache (7, 8) as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. After given the fundamental definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space (4), and intuitionistic fuzzy topological space (5, 6) to the case of neutrosophic sets. Possible application to superstrings and  

More Concise Algebraic Topology: Localization, Completion, and Model Categories

Abstract: With firm foundations dating only from the 1950s, algebraic topology is a relatively young area of mathematics. There are very few textbooks that treat fundamental topics beyond a first course, and many topics now essential to the field are not treated in any textbook. J. Peter May's "A Concise Course in Algebraic Topology" addresses the standard first course material, such as fundamental groups, covering spaces, the basics of homotopy theory, and homology and cohomology. In this sequel, May and his coauthor, Kathleen Ponto, cover topics that are essential for algebraic topologists and others interested in algebraic topology, but that are not treated in standard texts. They focus on the localization and completion of topological spaces, model categories, and Hopf algebras. The first half of the book sets out the basic theory of localization and completion of nilpotent spaces, using the most elementary treatment the authors know of. It makes no use of simplicial techniques or model categories, and it provides full details of other necessary preliminaries. With these topics as motivation, most of the second half of the book sets out the theory of model categories, which is the central organizing framework for homotopical algebra in general. Examples from topology and homological algebra are treated in parallel. A short last part develops the basic theory of bialgebras and Hopf algebras.

• on pre*- continuous and m-pre*continuous mappings

Abstract: The purpose of this paper is to introduce pre*continuous and M-pre*continuous functions in topological spaces and study some of their basic properties. Also we define pre*open, pre*closed, M-pre*open and M-pre*closed functions and investigate their relations among themselves.

Mapping topological order in coordinate space

Abstract: The organization of the electrons in the ground state is classified by means of topological invariants, defined as global properties of the wave function. Here we address the Chern number of a two-dimensional insulator and we show that the corresponding topological order can be mapped by means of a ``topological marker,'' defined in $\\mathbf{r}$ space, and which may vary in different regions of the same sample. Notably, this applies equally well to periodic and open boundary conditions. Simulations over a model Hamiltonian validate our theory.

Topological and Simplicial Models of Identity Types

Abstract: In this paper we construct new categorical models for the identity types of Martin-Lof type theory, in the categories Top of topological spaces and SSet of simplicial sets. We do so building on earlier work of Awodey and Warren [2009], which has suggested that a suitable environment for the interpretation of identity types should be a category equipped with a weak factorization system in the sense of Bousfield--Quillen. It turns out that this is not quite enough for a sound model, due to some subtle coherence issues concerned with stability under substitution; and so our first task is to introduce a slightly richer structure, which we call a homotopy-theoretic model of identity types, and to prove that this is sufficient for a sound interpretation.Now, although both Top and SSet are categories endowed with a weak factorization system---and indeed, an entire Quillen model structure---exhibiting the additional structure required for a homotopy-theoretic model is quite hard to do. However, the categories we are interested in share a number of common features, and abstracting these leads us to introduce the notion of a path object category. This is a relatively simple axiomatic framework, which is nonetheless sufficiently strong to allow the construction of homotopy-theoretic models. Now by exhibiting suitable path object structures on Top and SSet, we endow those categories with the structure of a homotopy-theoretic model and, in this way, obtain the desired topological and simplicial models of identity types.

On πgθ-Closed sets in Topological spaces

Abstract: In this paper a new class of sets called πgθ -closed is introduced and its properties are studied. Further the notion of πgθ-T½ space and πgθ -continuity are introduced. Mathematics Subject Classification: 54A05.

Topological structure of generalized rough sets

Abstract: In this paper, we investigate some properties of (X,@t"@q) such as compactness, separate property, Lindelof property and connectedness, where (X,@t"@q) is induced by a reflexive relation @q on X which may be infinite. Moreover, we introduce the concept of approximating spaces and research their characterizations and properties. Particularly, we obtain a sufficient and necessary condition that topological spaces are approximating spaces. These will be not only conducive to better understanding basic concepts and properties of a rough set, but also have theory and actual significance to topology.

Soft generalized closed sets in soft topological spaces

Abstract: In the present paper, we introduce soft generalized closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters. A sufficient condition for a soft g-closed set to be a soft closed is also presented. Moreover, the union and intersection of two soft g-closed sets are discussed. Finally, the new soft separation axiom, namely soft 1 2 T- space is introduced and its basic properties are investigated.

On soft topological space via semiopen and semiclosed soft sets

Abstract: This paper introduces semiopen and semiclosed soft sets in soft topological spaces. The notions of interior and closure are generalized using these sets. A detail study is carried out on properties of semiopen, semiclosed soft sets, semi interior and semi closure of a soft set in a soft topological space. Various forms of soft functions, like semicontinuous, irresolute, semiopen soft functions are introduced and characterized. Further soft semicompactness,soft semiconnectedness and soft semiseparation axioms are introduced and studied.

Covering maps for locally path-connected spaces

Abstract: We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow. If $X$ is path-connected, then every Peano covering map is equivalent to the projection $\widetilde X/H\to X$, where $H$ is a subgroup of the fundamental group of $X$ and $\widetilde X$ equipped with the basic topology. The projection $\widetilde X/H\to X$ is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on $\widetilde X$ for which one has a characterization of $\widetilde X/H\to X$ having the unique path lifting property if $H$ is a normal subgroup of $\pi_1(X)$. Namely, $H$ must be closed in $\pi_1(X)$. Such groups include $\pi(\mathcal{U},x_0)$ ($\mathcal{U}$ being an open cover of $X$) and the kernel of the natural homomorphism from the fundamental group to the Cech fundamental group.

Berge's Theorem for Noncompact Image Sets

Abstract: For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge's theorem states lower semi-continuity of the minimum of this function taken over the image sets. It assumes that the image sets are compact. For Hausdorff topological spaces, this paper extends Berge's theorem to set-valued mappings with possible noncompact image sets and studies relevant properties of minima.

On mixed fuzzy topological spaces and countability

Abstract: In this paper, we introduce a new type of mixed fuzzy topological space. We define countability on mixed fuzzy topological spaces. We investigate its different quasi type properties.

On extremally disconnected generalized topologies

Abstract: Extremal disconnectedness is further investigated for generalized topological spaces. It is found that extremally disconnected generalized topological spaces are a rich source of generalized lower semi-continuous and generalized upper semi-continuous mappings.

Skew category algebras associated with partially defined dynamical systems

Abstract: We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Top^op and show that it defines what we call a ske ...

C ∗ -algebras over topological spaces: filtrated k-theory

Abstract: We define the filtrated K-theory of a C � -algebra over a finite topo- logical space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C � -algebras over a space X with four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this space X, we enrich filtrated K-theory by another K-theory functor to a com- plete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

On some structures of soft topology

Abstract: In this paper, we define soft exterior and study its basic properties. We establish several important results relating soft interior, soft exterior, soft closure, and soft boundary in soft topological spaces. Moreover, we characterize soft open sets, soft closed sets, and soft clopen sets via soft boundary. All these findings will provide a base to researchers who want to work in the field of soft topology and will help to establish a general framework for applications in practical fields.

Monodromy Filtrations and the Topology of Tropical Varieties

Abstract: We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, “multiplicity-free” parameterization of Trop by a topological space and give a geometric interpretation of the cohomology of in terms of the action of a monodromy operator on the cohomology of . This gives bounds on the Betti numbers of in terms of the Betti numbers of which constrain the topology of Trop . We also obtain a description of the top power of the monodromy operator acting on middle cohomology of in terms of the volume pairing on .

On Generalized Pre Regular Weakly (gprw)-Closed Sets in Topological Spaces

Abstract: The aim of this paper is to introduce and study the new class of sets called gprw-closed sets. This new class of sets lies between the class of regular weakly closed (briefly rw-closed) sets and the class of generalized pre regular closed (briefly gpr-closed) sets. And also we study the fundamental properties of this class of sets.

Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension

Abstract: For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed k>1, there is a polynomial-time algorithm that, for a 1-connected topological space X given as a finite simplicial complex, or more generally, as a simplicial set with polynomial-time homology, computes the k-th homotopy group \pi_k(X), as well as the first k stages of a Postnikov system of X. Combined with results of an earlier paper, this yields a polynomial-time computation of [X,Y], i.e., all homotopy classes of continuous mappings X -> Y, under the assumption that Y is (k-1)-connected and dim X Y, and the question is the extendability of f to all of X. The algorithms are based on the notion of a simplicial set with polynomial-time homology, which is an enhancement of the notion of a simplicial set with effective homology developed earlier by Sergeraert and his co-workers. Our polynomial-time algorithms are obtained by showing that simplicial sets with polynomial-time homology are closed under various operations, most notably, Cartesian products, twisted Cartesian products, and classifying space. One of the key components is also polynomial-time homology for the Eilenberg--MacLane space K(Z,1), provided in another recent paper by Krcal, Matousek, and Sergeraert.