Search

We construct a counterexample that disproves the claim of Theorem 0.2 of the paper “Injective mappings and solvable vector fields of Euclidean spaces”, which appeared in Topology Appl. 136 (2004), 261–274. [ABSTRACT FROM AUTHOR]

In this paper, we give a classification of orientation reversing periodic maps on closed surfaces which generalizes the theory of Nielsen for the orientation preserving periodic maps. On one hand, we give a collection of data for each orientation reversing periodic map such that two periodic maps with the same data must be conjugate to each other. On the other hand, we give the criterion to judge when two different collections of data correspond to the same conjugacy class. As an application of the results of this paper, we shall show that a given orientation reversing periodic map on Σ g with period larger than or equal to 3 g must be conjugate to the power of a list of particular types of periodic maps. [ABSTRACT FROM AUTHOR]

The present paper establishes a homotopy based on Marcus–Wyse (for brevity M -) topology, which can contribute to the classification of 2D digital images in an M -topological approach. Since M -topology is considered in the Euclidean 2D space with integer coordinates, it can be used in studying 2D digital images from the viewpoints of both digital topology and digital geometry such as image processing, image analysis, computer graphics, mathematical morphology and so forth. To develop the homotopy, the present paper uses two maps such as an M -continuous map and Marcus–Wyse adjacency (for brevity MA -) map because they have their own features and merits. Besides, using this homotopy, the present paper proposes an MA -homotopy equivalence and further, MA -contractibility which can be also used in classifying digital images. Finally, the paper establishes an MA -homotopic thinning derived from the above homotopy, which can contribute to the compression of 2D digital images. [ABSTRACT FROM AUTHOR]

In the first half of the paper, we are concerned with the problems of existence (and searching) of coincidence points and the common preimage of a closed subset (in particular, a common root) in the case of a finite system of mappings of one metric space to another one. The second half of the paper is devoted to fiberwise variants of Arutyunov's theorem on coincidence points. Obtaining the main results of the paper is based on the use of the class of almost exactly ( α , β ) -search functionals that is wider than Fomenko's class of ( α , β ) -search functionals. [ABSTRACT FROM AUTHOR]

The present paper establishes two new maps such as an M -map and an M -isomorphism which are generalizations of a Marcus Wyse (for brevity, M -) continuous map and an M -homeomorphism because an M -continuous map is so rigid that some geometric transformations are not M -continuous maps (see Remark 3.2 ). Furthermore, it proves that in Z 2 an M -map and an M -isomorphism are equivalent to a (digitally) 4-continuous map and a (digitally) 4-isomorphism, respectively. Besides, the paper proves that SC MA l 1 is M -isomorphic to SC MA l 2 if and only if l 1 = l 2 , where SC MA l means a simple closed Marcus Wyse adjacent (for brevity, MA -) curve with l elements in Z 2 . Finally, the paper proves that MAC is equivalent to DTC ( 4 ) (see Theorem 6.7 ), where MAC is the category whose objects are M -topological spaces ( X , γ X ) with MA -adjacency and morphisms are all M -maps f : ( X , γ X ) → ( Y , γ Y ) for every ordered pair of objects ( X , γ X ) and ( Y , γ Y ) , and DTC ( 4 ) is the category whose objects are digital images ( X , k ) in Z 2 and morphisms are (digitally) 4-continuous maps. Besides, we propose the notion of an MA -retract for compressing 2D digital spaces. Using this new approach, we can substantially study and classify 2D digital topological spaces and 2D digital images. [ABSTRACT FROM AUTHOR]

Let X be a compact polyhedron that is homotopy equivalent to the figure-eight. This is a survey paper for the WYK algorithm which provides computations for the Nielsen number of self-maps of X . The WYK algorithm consists of three parts and each part was introduced separately by Wagner [11] , Yi and Kim [12] , and Kim [8] . In this paper, we present the whole algorithm so that people can more easily use the algorithm. [ABSTRACT FROM AUTHOR]

In the paper one considers the Hawaiian earring ( H , ⁎ ) , the wedge ( P , ⁎ ) of a sequence of 1-spheres and their Cartesian product ( H × P , ⁎ ) . One also considers the shape morphisms S [ π H ] , S [ π P ] , induced by the canonical projections π H : H × P → H , π P : H × P → P . The shape-theoretic problem asks if there exist a polyhedron Z and a shape morphism H : Z → H × P , H ≠ S [ ⁎ ] , such that S [ π H ] H = S [ ⁎ ] and S [ π P ] H = S [ ⁎ ] . Here S [ ⁎ ] denotes the shape morphisms, induced by the constant mappings ⁎ : Z → H × P , ⁎ : Z → H , and ⁎ : Z → P . Answering this problem affirmatively, would imply that the Cartesian product H × P is not a product in the shape category of topological spaces. The main result of the paper establishes equivalence between the shape-theoretic problem and a problem involving phantom mappings. [ABSTRACT FROM AUTHOR]

Abstract In [1] , Escardó, Lawson and Simpson introduced the concept of co-sober spaces and asked whether a sober space is co-sober. In this paper, we answer this question in the negative by a counterexample. The other main results are: (1) A closed subspace of a co-sober space is co-sober; (2) A saturated subspace of a co-sober space is co-sober; (3) If (X , τ) is a co-sober space and a continuous mapping c : (X , τ) → (X , τ) is a closure operator with respect to the specialization preorder of (X , τ) , then the subspace (c (X) , τ | c (X)) is a co-sober space; (4) If the Smyth power space P s (X) of a topological space X is co-sober, then X is co-sober. [ABSTRACT FROM AUTHOR]

Abstract In 1995 Heath, Keppelmann and Wong studied a Nielsen type number N F (f , p) for a fibre preserving map f of fibration p. This number is a lower bound for the least number of fixed points within the fibre homotopy class of f. In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers N P n F (f , p) and N Φ n F (f , p) for periodic points of a fibre preserving map. These numbers, which can be thought of as the dual of periodic Nielsen type numbers for a relative map due to Heath, Schirmer and You, are bigger than the ordinary Nielsen type numbers. The definition of N P n F (f , p) and N Φ n F (f , p) is reminiscent of the naive type addition formulae for periodic points of fibre preserving maps due to Heath and Keppelmann. Some calculations and examples are given. [ABSTRACT FROM AUTHOR]

Given a continuum X , let F n ( X ) denote the hyperspace of nonempty subsets of X with at most n points. For n ≥ 2 , let S F n ( X ) = F n ( X ) / F 1 ( X ) be the quotient space. Given a mapping between continua f : X → Y , we consider the induced mappings f n : F n ( X ) → F n ( Y ) and S f n : S F n ( X ) → S F n ( Y ) . Given a class of mappings M , in this paper we consider relations between the statements f ∈ M , f n ∈ M and S f n ∈ M , and we answer some questions about these relations considering the following classes of mappings: almost monotone, atriodic, freely decomposable and joining. [ABSTRACT FROM AUTHOR]

Let C ( X ) be the hyperspace of subcontinua of a continuum X . A Whitney block is a set of the form μ − 1 ( [ s , t ] ) , where μ : C ( X ) → [ 0 , 1 ] is a Whitney map and 0 ≤ s < t ≤ 1 . In this paper, we study the following implication: if X has property P , then each Whitney block in C ( X ) has property P . We consider the following properties: connectedness im kleinen, being absolute neighborhood retract, local contractibility, and m -mutual aposyndesis. [ABSTRACT FROM AUTHOR]

Let f 1 , . . . , f k : X → N be maps from a complex X to a compact manifold N , k ≥ 2 . In previous works [1,12] , a Lefschetz type theorem was established so that the non-vanishing of a Lefschetz type coincidence class L ( f 1 , . . . , f k ) implies the existence of a coincidence x ∈ X such that f 1 ( x ) = . . . = f k ( x ) . In this paper, we investigate the converse of the Lefschetz coincidence theorem for multiple maps. In particular, we study the obstruction to deforming the maps f 1 , . . . , f k to be coincidence free. We construct an example of two maps f 1 , f 2 : M → T from a sympletic 4-manifold M to the 2-torus T such that f 1 and f 2 cannot be homotopic to coincidence free maps but for any f : M → T , the maps f 1 , f 2 , f are deformable to be coincidence free. [ABSTRACT FROM AUTHOR]

The symbol S c ( X ) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X . This hyperspace is endowed with the Vietoris topology. For a given mapping between Hausdorff spaces f : X → Y , define the induced mapping S c ( f ) by S c ( f ) ( A ) = f [ A ] (the image of A under f ) for every A ∈ S c ( X ) . In the current paper, we study under which conditions the fact that f belongs to a given class of mappings M implies that S c ( f ) belongs to M , and vice versa. [ABSTRACT FROM AUTHOR]

For a Tychonoff space X , let ↓ C F ( X ) denote the collection of the hypographs of all continuous maps from X to [ 0 , 1 ] with the Fell topology. We show that, for a Tychonoff non-discrete k-space X , the function space ↓ C F ( X ) is homeomorphic to c 0 ∪ ( Q ∖ Σ ) if ↓ C F ( X ) is metrizable and the set of isolated points of X is dense in X , where Q = [ − 1 , 1 ] N is the Hilbert cube, Σ = { ( x n ) ∈ Q : sup ⁡ | x n | < 1 } and c 0 = { ( x n ) ∈ Σ : lim ⁡ x n = 0 } are its subspaces. Combining results in the previous papers of the series, we give the topological classification for all metrizable function spaces ↓ C F ( X ) of k-spaces X . [ABSTRACT FROM AUTHOR]

It is obtained a selection theorem for paraconvex-valued mappings possessing two natural properties. Several known results follow easily from this theorem. Moreover, it implies the affirmative solution to a question stated in a paper of N.R. Loufouma Makala. [ABSTRACT FROM AUTHOR]

This paper continues the study of some movability and co-movability properties for shape morphisms and maps introduced by the authors in [9] . For such morphisms the properties of movability and co-movability with respect to a subcategory are introduced. In particular the properties of n -movability and n -co-movability for maps are defined and studied. As applications, some implications of these properties on the homology and homotopy pro-groups and shape groups are proved. [ABSTRACT FROM AUTHOR]

In the paper it is proved that there exists a continuous surjective mapping F : X → Y , where X and Y are separable complete metric spaces of transfinite dimension ind less than or equal to α d ∈ ω + ∪ { ∞ } and α r ∈ ω + ∪ { ∞ } , respectively, such that, for each continuous surjective mapping f : X → Y , where X and Y are compact metric spaces of transfinite dimension ind less than or equal to α d and α r , respectively, there are isometries i : X → X and j : Y → Y satisfying the relation F ∘ i = j ∘ f . This result remains true if we suppose that, for each mapping f : X → Y , Y coincides with a fixed separable complete metric space Y and that the corresponding mapping j : Y → Y is the identity. [ABSTRACT FROM AUTHOR]

In this paper we provide some affirmative results and some counterexamples for a solution of the splitting problem for n multivalued mappings, n > 2 . [ABSTRACT FROM AUTHOR]

The present paper can be regarded as a natural, significant, multivalued extension of some results matching the general periodic point theory (like the celebrated Sharkovsky-type theorems) and the standard Nielsen fixed point theory. Concretely, the coexistence of irreducible orbits of coincidences is established for multivalued circle maps by means of Nielsen-type topological invariants. A well known theorem for single-valued maps, obtained independently by Efremova [1] and Block et al. [2] , is nontrivially generalized in this way. Some further possibilities for admissible maps on tori are indicated. Several illustrative examples are supplied. The crucial idea is based on detecting the kind of a complete isomorphism between periodic points of associated single-valued maps and irreducible orbits of coincidences of given multivalued admissible maps on tori. [ABSTRACT FROM AUTHOR]

Spaces of order arcs have been studied by several authors [1,4,9] , but mappings between them have not been considered. In this paper we focus on the spaces of maximal order arcs and introduce the concept of induced maps between those spaces. We show several relationships between properties of a map and those of its induced map. [ABSTRACT FROM AUTHOR]

In this paper, we study the Lusternik–Schnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Several properties of this new invariant are shown, including its relevance to simplicial products and fibrations. We relate this category of a map to the classical Lusternik–Schnirelmann category of a map between finite topological spaces. Finally, we show how the simplicial category of a map may be used to define and study a simplicial version of the category weight of Y. Rudyak and J. Strom. [ABSTRACT FROM AUTHOR]

Given a continuum X and a positive integer n , let F n ( X ) be the hyperspace of all nonempty subsets of X having at most n points. Given a mapping f : X → Y between continua, we study the induced mapping f n : F n ( X ) → F n ( Y ) given by f n ( A ) = f ( A ) (the image of A under f ). In this paper, we prove some relationships among the mappings f and f n for the following classes of mappings: almost open, almost monotone, atriodic, feebly monotone, local homeomorphism, locally confluent, locally weakly confluent, strongly monotone and weakly semi-confluent. [ABSTRACT FROM AUTHOR]

This paper investigates the space C k ( ω ⁎ , ω ⁎ ) , the space of continuous self-maps on the Stone–Čech remainder of the integers, ω ⁎ , equipped with the compact-open topology. Our main results are that • C k ( ω ⁎ , ω ⁎ ) is Baire, • Stone–Čech extensions of injective maps on ω form a dense set of weak P -points in C k ( ω ⁎ , ω ⁎ ) , • it is independent of ZFC whether C k ( ω ⁎ , ω ⁎ ) contains P -points, and that • C k ( ω ⁎ , ω ⁎ ) is not an F -space, but contains, as ω ⁎ , no non-trivial convergent sequences. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of points u ∈ A N that are such that the omega-limit set ω ( u , σ ) is precisely X , where X ⊆ A N is an internally transitive shift space. We then use those results to study the possible structures of the omega-limit set of the turning point for a unimodal map. Examples are provided of unimodal maps f where no iterate of the turning point c is recurrent and ω ( c , f ) is either a minimal Cantor set or properly contains a minimal Cantor set. [ABSTRACT FROM AUTHOR]

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results. [ABSTRACT FROM AUTHOR]

In the paper the concept of covering (regularity) for set-valued mappings in partially ordered spaces is introduced. The coincidence points problem for set-valued mappings in partially ordered spaces is considered. Sufficient conditions for the existence of coincidence points of isotone and orderly covering set-valued mappings are obtained. It is shown that the known theorems on coincidence points of covering and Lipschitz mappings in metric spaces can be deduced from the obtained results. [ABSTRACT FROM AUTHOR]

In [9] , two new kinds of topologies called the open-point topology and the bi-point-open topology on C ( X ) , the set of all real-valued continuous functions on a Tychonoff space X , have been introduced and their separation and countability properties have been studied. In the present paper, we study the submetrizability and cardinal functions such as extent, cellularity, weight, pseudocharacter, character and tightness of the spaces C ( X ) equipped with the open-point and bi-point-open topologies. [ABSTRACT FROM AUTHOR]

For an ordered topological vector space Y and a , b ∈ Y , we write a ≪ b if b − a is an interior point of the positive cone. Modifying the earlier results of Borwein–Théra, in this paper, ≪ is extended over Y • • = Y ∪ { ∞ } ∪ { − ∞ } and a natural topology on Y • • is introduced. For a topological space X , and a non-trivial separable ordered topological vector space Y with an interior point of the positive cone, we show the following: X is normal and countably paracompact if and only if for every lower semi-continuous map f : X → Y • • and every upper semi-continuous map g : X → Y • • with g ≪ f , there exists a continuous map h : X → Y such that g ≪ h ≪ f . [ABSTRACT FROM AUTHOR]

In this paper, we prove that an example by Mysior has no regular G δ -diagonal, which can be used to show that open ≤2-to-one maps need not preserve regular G δ -diagonals or zero-set diagonals. We also discuss the diagonal of the Mrówka space. Let A be a maximal almost disjoint family on N , then for any countable subset B of A , the Mrówka space ψ ( A ∖ B ) has no regular G δ -diagonal. [ABSTRACT FROM AUTHOR]

In [9] , we have proved that every Nielsen zeta function for a self-map on an infra-nilmanifold is rational. In this paper, we will list all possible self-maps on infra-nilmanifolds up to dimension three and we will give the Nielsen zeta function for any of those maps, by using the techniques developed in [9] . [ABSTRACT FROM AUTHOR]

Let f : M → M be a self-map on a three-dimensional manifold M with S 3 -geometry. In this paper, we determine the possible values of N ( f ) where N ( f ) is the Nielsen number of f . [ABSTRACT FROM AUTHOR]

This paper gives several conditions sufficient for a topological group to be locally connected, locally path connected, or almost locally path connected. Structural theorems for non-locally connected groups arise, such as a limitation on the types of subspaces that can occur as somewhere dense subsets, and the meagerness of conjugacy classes in many instances. Improvements are given to certain open, or almost open, mapping theorems. New observations are made about the associated locally path connected topology of a topological group. [ABSTRACT FROM AUTHOR]

In this paper we prove that in the classes of all regular frames and all completely regular frames there are universal elements giving an affirmative answer to Problems 1.4 (1, 2) of [1] . To obtain these results we simplify the method of construction of universal elements in the class of all frames given in [1] and then use it for the considered classes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish relationships between Lebesque and inductive dimensions ( m , n ) -dim and ( m , n ) -Ind. The main results are the following: 1. For every normal space X we have ( m , n ) - dim X = ⌊ dim X n ⌋ . 2. For every metrizable space X we have ( m , n ) - dim X = ( m , n ) - Ind X . [ABSTRACT FROM AUTHOR]

We study the behaviour of fibre maps under exponentiation, i.e. given a fibration p : E → B we ask for which spaces X is the induced map between mapping spaces p ⁎ : E X → B X also a fibration. If X is a locally compact space, the positive answer follows easily by the exponential law so in this paper we consider more general spaces and show that the preservation of fibrations is related to the local homotopy properties of the space X . For example, if p is a Dold fibration and X admits a deformation retraction on a compactly generated space, then the induced map p ⁎ is also a Dold fibration. Similar results hold for Hurewicz fibrations with unique path-lifting and for covering spaces, and can be furthermore extended to spaces that admit some numerable cover, whose elements preserve fibration property. [ABSTRACT FROM AUTHOR]

In this paper, we study extensively the compact-Gδ-open topology on the set C (X) of all real-valued continuous functions on a Tychonoff space X. In particular, we study the induced map, submetrizability, completeness properties, countability properties, and nuclear and reflexive properties of the compact-Gδ-open topology on C (X). [ABSTRACT FROM AUTHOR]

In this paper we complete the attempt of H. Lefmann to show that Borel equivalence relations on the n-element subsets of 2ω, that respect an order type, have a finite Ramsey basis. [ABSTRACT FROM AUTHOR]