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A space X is called a k R -space, if X is Tychonoff and the necessary and sufficient condition for a real-valued function f on X to be continuous is that the restriction of f to each compact subset is continuous. In this paper, we mainly discuss the k R -property in the free topological groups, and generalize some well-known results of K. Yamada.

If X is a topological space and A X, then the number of distinct sets that can be obtained from A by using all possible compositions for operators i , c (where = ; ; ; ) introduced by Cs asz ar is at the most 25. Explicit expressions for these sets are provided. An example is provided where all the 25 di erent sets are determined. The result is also discussed for special cases such as when the space is extremally disconnected, resolvable, open-unresolvable, and partition spaces.

Using the cone theory and the lattice structure, we establish some methods of computation of the topological degree for the nonlinear operators which are not assumed to be cone mappings. As applications, existence results of nontrivial solutions for singular Sturm-Liouville problems are given. The nonlinearity in the equations can take negative values and may be unbounded from below.

Consider an arbitrary Hilbert space H endowed with a continuous product which induces a grading on H with respect to an abelian group G. We show that such a space H has the form H = cl ( U + ∑ j I j ) with U a closed subspace of H 1 (the factor associated to the unit element in G), and any I j a well described closed graded ideal of H , satisfying I j I k = 0 if j ≠ k . Under certain conditions, the graded simplicity of H is characterized and it is shown that H is the closure of the orthogonal direct sum of the family of its minimal (closed) graded ideals, each one being a graded simple graded Hilbert space.

Summary. In this paper we present selected properties of barycentric coordinates in the Euclidean topological space. We prove the topological correspondence between a subset of an ane closed space of E n and the set of vectors

This paper takes some investigations on generalized topological spaces with some µ-separations. Some characterizations of µTi-spaces for i = 0, 1, 2, 3, 4, µTD-spaces and µR0-spaces are obtained and some relations among these spaces are established.

In this article we study the existence of non-continuous linear functionals on topological vector spaces. Both sufficient and necessary conditions for the existence of such maps are found.

We give topological invariants for a wide class of abso-lutely isolated singularities of three-dimensional real vector fields. Our invariants are complete once the desingularization morphism $\pi$ is fixed. They are obtained in terms of a finite set of configurations depending only on the eigenvalues of the singularities.

The space of periodic Boehmians with ∆–convergence is a complete topological algebra which is not locally convex. A family of Boehmians {Tλ} such that T0 is the identity and Tλ1+λ2 = Tλ1 ∗Tλ2 for all real numbers λ1 and λ2 is called a one-parameter group of Boehmians. We show that if {Tλ} is strongly continuous at zero, then {Tλ} has an exponential representation. We also obtain some results concerning the infinitesimal generator for {Tλ}.

In this paper, relations between topological entropy, volume growth and the growth in topological complexity from homotopical and homological point of view are discussed for random dynamical systems. It is shown that, under certain conditions, the volume growth, the growth in fundamental group and the growth in homological group are all bounded from above by the topological entropy.

Let (X, d) be a compact metric space and let f : X → X be continuous. Let K ( X ) be the family of compact subsets of X endowed with the Hausdorff metric and define the extension f ¯ : K ( X ) → K ( X ) by f ¯ ( K ) = f ( K ) for any K ∈ K ( X ) . We prove that the topological entropy of f ¯ is greater or equal than the topological entropy of f, and this inequality can be strict. On the other hand, we prove that the topological entropy of f is positive if and only if the topological entropy of f ¯ is also positive.

A generalization is considered of Williams's well-known model of the attractor in the Lorenz system, the inverse limit of semiflows on branched manifolds that are suspensions over a discontinuous expanding map of a closed line interval. The generalization consists in the consideration of maps with several, rather than one, discontinuity points. A cardinal-valued topological invariant L-manuscript is constructed, which distinguishes a continuum of non-homeomorphic generalized models. A topological invariant distinguishing a continuum of non-homeomorphic geometric Lorenz attractors is obtained as a consequence.

A short introduction on topological properties of (regular and random) geometrical sets is presented along with some recent results concerning the behaviour of the Euler–Poincare characteristic with respect to the (Fortuin–Kasteleyn) random cluster measure.

In this work, some new generalized R-KKM type theorems for generalized R-KKM mappings with compactly open values are established in topological spaces without any convexity assumptions. As applications, a Ky Fan type matching theorem, fixed point theorem and coincidence theorem are obtained in topological spaces. These results generalize some known results from the recent literature.

In this paper generalized vector fields are studied. The notion of left invariant vector fields is extended by the use of top spaces. As a result a new kind of dynamics is extrapolated. By the topological properties of topological generalized groups, topological complete semi-dynamical systems are studied. A method for constructing a complete semi-dynamical system on a generalized coset space is deduced. An application in the genetic space and the whether system axe considered.

We consider m-fields that are generalizations of the Morse–Smale vector fields for manifolds with boundary. We construct complete topological invariants of m-fields on surfaces and m-fields without closed trajectories on three-dimensional manifolds. We also prove criteria for the topological equivalence of m-fields.

We study the operator equation AX = Y, where the operators X and Y are given and the operator A is required to lie in some von Neumann algebra. We derive a necessary and sufficient condition for the existence of a solution A. The condition is that there must exist a constant K so that, for all finite collections of operators {D 1 , D 2 ,..., D n } in the commutant, and all collections of vectors {f 1 , f 2 ,..., f n }, we have ∥Σ n j=1 D j Yf j ∥ ≤ K ∥ Σ n j=1 D j X f j ∥. We also study the equality ∥DY f ∥ = K∥DX f ∥, in connection with solving the equation AX = Y where the operator A is required to lie in some CSL algebra.

We prove that the invariants of the topological semiconjugation of Lorenz maps with β-transformations remains constant on the renormalization archipelagoes and analyze how the dynamics on the archipelagoes depends on its structure.

We study topological properties of orthogonal Hilbert-space valued linear mappings on von Neumann algebras (orthogonal vector fields). Also we consider a possibility of extension of an orthogonal vector field to the space of square integrable bilinear forms.

In this paper, the authors prove that the flows of homogeneous vector field Q(x) at infinity are topologically equivalent to the flows of the tangent vector field QT(u) (u ∈ S2) on the sphere S2, and show the theorems for the global topological classification of Q(x). They derive the necessary and sufficient conditions for the global asymptotic stability and the boundedness of vector field Q(x), and obtain the criterion for the global topological equivalence of two homogeneous vector fields.

It is proved that the arbitrary nondegenerate system in a linear complete topological space has a correspondence complete topological space of coefficients with canonical basis. Basicity criterion for systems in such spaces is given in terms of coefficient operator.

The Temperley-Lieb algebra is used to find a base for the vector space that is associated to a closed surface by the Topological Quantum Field Theory corresponding to the original Jones polynomial invariant

The Smirnov Class N+ of analytic functions on the disk has traditionally been studied with the complete metric topology given by ρ(f,g)= sup0

In 1992, Ramadan introduced the concept of a smooth topological space and relativeness between smooth topological space and fuzzy topological space in Chang's (1968) view points. In this paper we give a new definition of smooth topological space. This definition can be considered as a generalization of the smooth topological space which was given by Ramadan. Some general properties such as relative smooth continuity and relative smooth compactness are studied.

We consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these equivalences., To appear in Geometriae Dedicata, 23 pages, 8 figures, see also http://www.math.columbia.edu/~clein

We consider hyperbolic sets of saddle type for holomorphic mappings in P. Our main result relates topological entropy on such sets to a normal families condition on local unstable manifolds.

For various classes K K of topological spaces we prove that if X 1 , X 2 ∈ K {X_1},{X_2} \in K and X 1 , X 2 {X_1},{X_2} have isomorphic homeomorphism groups, then X 1 {X_1} and X 2 {X_2} are homeomorphic. Let G G denote a subgroup of the group of homeomorphisms H ( X ) H(X) of a topological space X X . A class K K of ⟨ X , G ⟩ \langle X,G\rangle ’s is faithful if for every ⟨ X 1 , G 1 ⟩ , ⟨ X 2 , G 2 ⟩ ∈ K \langle {X_1},{G_1}\rangle ,\langle {X_2},{G_2}\rangle \in K , if φ : G 1 → G 2 \varphi :{G_1} \to {G_2} is a group isomorphism, then there is a homeomorphism h h between X 1 {X_1} and X 2 {X_2} such that for every g ∈ G 1 φ ( g ) = h g h − 1 g \in {G_1}\;\varphi (g) = hg{h^{ - 1}} . Theorem 1: The following class is faithful: { ⟨ X , H ( X ) ⟩ | ( X \{ \langle X,H(X)\rangle |(X is a locally finite-dimensional polyhedron in the metric or coherent topology or X X is a Euclidean manifold with boundary) and for every x ∈ X x x \in X\;x is an accumulation point of { g ( x ) | g ∈ H ( X ) } } ∪ { ⟨ X , G ⟩ | X \{ g(x)|g \in H(X)\} \} \cup \{ \langle X,G\rangle |X is a differentiable or a P L PL -manifold and G G contains the group of differentiable or piecewise linear homeomorphisms } \} ∪ { ⟨ X , H ( X ) ⟩ | X \cup \{ \langle X,H(X)\rangle |X is a manifold over a normed vector space over an ordered field } \} . This answers a question of Whittaker [ W ] [{\text {W}}] , who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful: { ⟨ X , G ⟩ | X \{ \langle X,G\rangle |X is a locally compact Hausdorff space and for every open T ⊆ X T \subseteq X and x ∈ T { g ( x ) | g ∈ H ( X ) x \in T\;\{ g(x)|g \in H(X) and g ↾ ( X − T ) = Id } g \upharpoonright (X - T) = \operatorname {Id}\} is somewhere dense } \} . Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful: { ⟨ X , H ( X ) ⟩ | \{ \langle X,H(X)\rangle | (1) X X is a 0 0 -dimensional Hausdorff space; (2) for every x ∈ X x \in X there is a regular open set whose boundary is { x } \{ x\} ; (3) for every x ∈ X x \in X there are g 1 , g 2 ∈ G {g_{1,}}{g_2} \in G such that x ≠ g 1 ( x ) ≠ g 2 ( x ) ≠ x x \ne {g_1}(x) \ne {g_2}(x) \ne x , and (4) for every nonempty open V ⊆ X V \subseteq X there is g ∈ H ( X ) − { Id } g \in H(X) - \{ \operatorname {Id}\} such that g ↾ ( X − V ) = Id } g \upharpoonright (X - V) = \operatorname {Id}\} . Note that (2) is satisfied by 0 0 -dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from R {\mathbf {R}} . Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class K T {K^T} of trees { ⟨ Aut ⁡ ( T ) , T ; ≤ , ∘ , Op ⟩ | T ∈ K T } \{ \langle \operatorname {Aut}(T),T; \leq , \circ ,\operatorname {Op}\rangle |T \in {K^T}\} is first-order interpretable in { Aut ⁡ ( T ) | T ∈ K T } \{ \operatorname {Aut}(T)|T \in {K^T}\} .

Convergence spaces are studied from a geometrical point of view. In many cases we may construct a topological space VirtX, which can be considered as the resolution of the topological singularities ofX.

A theorem about calculation of topological degree for set-contraotions is presented, and then eigenvectors and nonzero fixed points of strict-set-contractions are inveatigated.

LetG denote an infinite,σ-compact, locally compact topological group. In this paper a construction is given for a topological transformation groupHG with the Hilbert spaceL 2 (G × G) as a phase space such that any topological transformation group (G, X, π) can be embedded inH G , providedX is a separable metrizable space andπ is a bounded action. The class of such topological transformation groups contains all actions ofG on separable, metrizable, locally compact spaces.

In this paper we give a complete topological classification for multiple saddle-connections of a real analytic vector field along an axis Γ in an ambient space of dimension three, under the assumption that Γ is the intersection of two invariant surfaces D 1 and D 2 .

In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679–690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results.