A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

This paper first presents a characterization of three classes of negligible closed convex sets (i.e., Gauss null sets, Aronszajn null sets and cube null sets) in terms of non-support points; then gives a generalization of Gâteaux differentiability theorems of Lipschitz mapping from open sets to those closed convex sets admitting non-support points; and as their application, finally shows that a closed convex set in a separable Banach space X can be Lipschitz embedded into a Banach space Y with the Radon–Nikodym property if and only if the closure of its linear span is linearly isomorphic to a closed subspace of Y.     

Regular-closed Functions and Hausdorff Spaces

A function f: X → Y is said to be regular-closed [4] if for each regular closed set F of X, f(F) is closed in Y. It is shown that let f: X → Y be a regular-closed surjection, then Y is HAUSDORFF if either f is open or X is normal.     

Polynomial hulls and H￥H^\infty control for a hypoconvex constraint

We say that a subset of is hypoconvex if its complement is the union of complex hyperplanes. Let be the closed unit disk in , . We prove two conjectures of Helton and Marshall. Let be a smooth function on whose sublevel sets have compact hypoconvex fibers over . Then, with some restrictions on , if Y is the set where is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the infimum is attained by a unique bounded analytic f which in fact is also smooth on . We also prove that if varies smoothly with respect to a parameter, so does the unique f just found. Received: 18 December 1998 / Published online: 28 June 2000     

Factorization of weakly continuous differentiable mappings

Given real Banach spaces X and Y, let C _wbu¹(X, Y) be the space, introduced by R.M. Aron and J.B. Prolla, of C ¹ mappings from X into Y such that the mappings and their derivatives are weakly uniformly continuous on bounded sets. We show that f ∈ C _wbu¹(X, Y) if and only if f may be written in the form f = g ∘ S, where the intermediate space is normed, S is a precompact operator, and g is a Gateaux differentiable mapping with some additional properties.     

Closed maps on metric spaces

The following result is proved: Let Y be the image of a metric space X under a closed map f. Then every ?f^-1(y) is Lindelöf if and only if Y has a point-countable k-network.     

The structure of LC-continuous functions

A function f is LC-continuous if the inverse image of any open set is a locally closed set; i.e., an intersection of an open set and a closed set. The aim of this paper is to prove the following theorem: Let f: X→Y be an LC-continuous function onto a separable metric space Y. Then X can be covered by countably many subsets T _n ⊂X such that each restriction f∣T _n is continuous at all points of T _n .     

On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

We investigate the existence of a separately continuous function f: X × Y → ℝ with a one-point set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x ₀ ∈ X and y ₀ ∈ Y, a separately continuous function f: X × Y → ℝ with the set of discontinuity points {(x ₀, y ₀)} exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x ₀ and y ₀, respectively.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 94–101, January, 2005.     

On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE ^w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE ^w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.     

Points of joint continuity and large oscillations

For topological spaces X and Y and a metric space Z, we introduce a new class N( X ×Y, Z ) \mathcal{N}\left( {X \times Y,\,Z} \right) of mappings f: X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping f from this class and any countable-type set B in Y, the set C _B (f) of all points x from X such that f is jointly continuous at any point of the set {x} × B is residual in X: We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and f ? N( X ×Y, Z ) f \in \mathcal{N}\left( {X \times Y,\,Z} \right) , then, for any ε > 0, the projection of the set D ^ε (f) of all points p ∈ X × Y at which the oscillation ω _f (p) ≥ ε onto X is a closed set nowhere dense in X.     

Epsilon Nielsen coincidence theory

We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ^∈ (f, g) to be the minimum number of coincidence points of any maps f ₁ and g ₁ such that f ₁ is ∈ ₁-homotopic to f, g ₁ is ∈ ₂-homotopic to g and ∈ ₁ + ∈ ₂ < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ^∈ (f, g) moving only one rather than both of the maps. In particular, if X = Y is a closed Riemannian manifold and id _Y is its identity map, then µ ^∈ (f, id _Y ) is equal to the ∈-minimum fixed point number of f defined by Brown. If X and Y are orientable closed Riemannian manifolds of the same dimension, we define an ∈-Nielsen coincidence number N ^∈ (f, g) as a lower bound for µ ^∈ (f, g). Our constructions and main results lead to an epsilon root theory and we prove a Minimum Theorem in this special approach.     

Cutting and Pasting of Elliptic Operators

If A : C∞E → C∞F is an elliptic operator between sections of vector bundles E, F over a closed smooth n-manifold X, Y a smooth (n – 1)-submanifold of X with trivial normal bundle, and Ψ = (ΨE, ΨF) a pair of automorphisms of E | Y and F | Y inducing a diffeomorphism f of Y and commuting with the principal symbol σ A of A over Y, then an elliptic operator A^Ψ is (uniquely up to operators of lower order) defined between sections of vector bundles E^Ψ, F^Ψ over a closed manifold X^f, all obtained by cutting and pasting the respective objects along Y. The difference index A^Ψ – index A is investigated and the relations with additivity properties of topological invariants and with classical transmission problems are explained.     

Preparation of papers

LetF be a family of real-valued maps onR ⁿ, and letY be a subset ofR ⁿ. Denote byS(Y|F) the set of ally* Y such that, for somef F,f(y)f(y*) for ally inY. Let us say thatF is a scalarization family if, for any subsetY,S(Y|F) is equal to the set of properly efficient points inY. General conditions forF to be a scalarization family were given in Ref. 1. However, scalarization families must contain nondifferentiable functions. In this note, it is shown that, if the condition of Ref. 1 which forces nondifferentiability is dropped, thenS(Y|F) is dense in the set of properly efficient points.     

On the Monodromy of a Multivalued Function along Its Ramification Locus

We consider multivalued analytic functions in ⁿ) whose set of singular points occupies a very small part of ⁿ). Under a mapping of a topological space Y into ⁿ), such a function f can induce a multivalued function on Y. This is possible even if the image of Y entirely lies in the ramification set of f. We estimate the monodromy group of the induced function via the monodromy group of f.     

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2^Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ^* ε K such that f g(χ^*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.     

Refinable maps on ANR's

The map F:X→Y is refinable if for each >0 there is an -map f from X onto Y that is -close to F. The closed set A in X is N-elementary if each neighborhood U of A contains a neighborhood V such that the natural homomorphism ^N(U) → ^N(V) has finitely generated image. If X is a compact ANR, then every closed subset is N-elementary for every N.Suppose F:X → Y is a refinable map between compacta. Then:If B is a compactum in Y such that F^-1B is N-elementary in X then F induces an isomorphism from ^N(B) to ^N(F^-1B). In particular, if X is an ANR, then F induces isomorphisms ^N(B) ^N (F^-1B) and _N(F^-1B) _N(B).If X=S³ and Y=S³/A, then A is cellular.If X is a finite-dimensional ANR, then Y is an ANR if one of the following is true: (1) Y is LC¹, (2) F^-1(y) is locally connected for each y Y, (3) F^-1(y) is approximately 1-connected for each y Y or (4) for each >0 the f in the above definition can be chosen to be monotone.Applications are also made to generalized manifolds and ANR's in 2-dimensional manifolds.     

树映射的链回归点与拓扑熵

Let f be a tree map,P(f) the set of periodic points of f and CR(f) the set of chain recurrent points of f. In this paper,the notion of division for invariant closed subsets of a tree map is introduced. It is proved that: (1) fhas zero topological entropy if and only if for any x∈CR(f)-P(f) and each natural number s the orbit of x under f^5 has a division; (2) If f has zero topological entropy,then for any xECR(f)--P(f) the w-limit set of x is an infinite minimal set.     

Smooth extensions of functions on separable Banach spaces

Let X be a Banach space with a separable dual X*. Let ${Y\subset X}Let X be a Banach space with a separable dual X*. Let Y ì X{Y\subset X} be a closed subspace, and f:Y?\mathbbR{f:Y\rightarrow\mathbb{R}} a C ¹-smooth function. Then we show there is a C ¹ extension of f to X.     

Connected components of attractors and other stable sets

Summary We study the action of a continuous mapf on a locally connected metric space, with an emphasis on the action induced byf on the space of connected components of an attractorA or a stable setY off. The main result for attractors is that, ifA is an attractor in the sense of C. Conley, thenf permutes the components ofA and that the permutation is cyclic if and only if the attractor is indecomposable. This is related to similar results obtained recently by Buescu and Stewart. In the case of an indecomposable stable invariant setY we show that either the action off on the components ofY is as described above for the action on the attractor, or elseY has infinitely many components, the action off on them is a generalized adding machine, so that there are no periodic points inY. Results along these lines are due to Buescu and Stewart and to Melbourne, Dellnitz and Golubitsky.     

An implicit function theorem for algebraically closed fields

Let F be an algebraically closed field of characteristic 0, and let f : F ⁿ → F. We show that f is implicitly definable by a system of polynomial equations if and only if f is a special kind of piecewise rational function.