On properties of h-homogeneous spaces of first category

A metric space X is called h-homogeneous if and each nonempty open-closed subset of X is homeomorphic to X. We describe how to assign an h-homogeneous space of first category and of weight k to any strongly zero-dimensional metric space of weight ?k. We investigate the properties of such spaces. We show that if Q is the space of rational numbers and Y is a strongly zero-dimensional metric space, then Q×Yω is an h-homogeneous space and F×Q×Yω is homeomorphic to Q×Yω for any Fσ-subset F of Q×Yω. L. Keldysh proved that any two canonical elements of the Borel class α are homeomorphic. The last theorem is generalized for the nonseparable case.     

Spaces of maps into topological group with the Whitney topology

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps endowed with the Whitney (graph) topology and by Cc(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l₂-manifold. In this article we show that if X is non-compact and not end-discrete then Cc(X,G) is an (R^∞×l₂)-manifold, and moreover the pair (C(X,G),Cc(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l₂.     

Distances from selectors to spaces of Baire one functions

Given a metric space X and a Banach space (E,‖⋅‖) we study distances from the set of selectors Sel(F) of a set-valued map to the space B₁(X,E) of Baire one functions from X into E. For this we introduce the d-τ-semioscillation of a set-valued map with values in a topological space (Y,τ) also endowed with a metric d. Being more precise we obtain that

     

Computability of a topological poset

For subspaces X and Y of Q the notation Xh_?Y means that X is homeomorphic to a subspace of Y and X∼Y means Xh_?Yh_?X. The resulting set P(Q)/∼ of equivalence classes is partially-ordered by the relation if Xh_?Y. In a previous paper by the author it was established that this poset is essentially determined by considering only the scattered X⊆Q of finite Cantor-Bendixson rank. Results from that paper are extended to show that this poset is computable.     

Countably compact hyperspaces and Frolík sums

Let H⁰(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u∈ω^∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction.We define the property R(κ): for every family of closed subsets of X separated by pairwise disjoint open sets and any family of natural numbers, the product is countably compact, and prove that if H(X) is countably compact for a T₂-space X then X satisfies R(κ) for all κ. A space has R(1) iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg: if X is T₂ and H(X) is countably compact, then so is Xn for all n<ω. We also prove that, for κ<t, if the T₃ space X satisfies a weak form of R(κ), the orbit of every point in X is dense, and X contains κ pairwise disjoint open sets, then Xκ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T₃, homogeneous, and H(X) is countably compact, then so is Xω.Then we study the Frolík sum (also called “one-point countable-compactification”) of a family . We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product _∏α<κH⁰(Xα) embeds into .     

Cells in hyperspaces

Let C(X) denote the hyperspace of subcontinua of a continuum X. For A∈C(X), define the hyperspace . Let k∈N, k?2. We prove that A is contained in the core of a k-od if and only if C(A,X) contains a k-cell.     

Absolutely summing polynomials on Banach spaces with unconditional basis

If X is a Banach space with a normalized unconditional Schauder basis (xn), we define whenever and obtain estimates for μX,(xn) when every continuous m-homogeneous polynomial from X into Y is absolutely (q,1) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m=1, we obtain new contributions to the linear theory of absolutely summing operators.     

Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb^′(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever g○f=0 for any f∈Cb(X,E), g∈E^′.     

A local criterion for Weyl modules for groups of type A

Let G be a universal Chevalley group over an algebraically closed field and U⁻ be the subalgebra of generated by all divided powers X_α,m with α<0. We conjecture an algorithm to determine if , where F∈U⁻, ω is a dominant weight and is a highest weight vector of the Weyl module Δ(ω). This algorithm does not use bases of Δ(ω) and is similar to the algorithm for irreducible modules that involves stepwise raising the vector under investigation. For an arbitrary G, this conjecture is proved in one direction and for G of type A in both.     

Codimension reduction and second fundamental form of CR submanifolds in complex space forms

Considering n-dimensional real submanifolds M of a complex space form which are CR submanifolds of CR dimension , we study the condition h(FX,Y)+h(X,FY)=0 on the structure tensor F naturally induced from the almost complex structure J of the ambient manifold and on the second fundamental form h of submanifolds M.     

Open selections on smooth fans

Given a dendroid X, an open selection is an open map such that s(A)∈A for every A∈C(X). We show that a smooth fan X admits an open selection if and only if X is locally connected.     

The bicompletion of the Hausdorff quasi-uniformity

We study conditions under which the Hausdorff quasi-uniformity UH of a quasi-uniform space (X,U) on the set P₀(X) of the nonempty subsets of X is bicomplete.Indeed we present an explicit method to construct the bicompletion of the T₀-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform T₀-spaces (X,U) for which the Hausdorff quasi-uniformity of their bicompletion on is bicomplete.     

Retractions and contractibility in hyperspaces

Given a metric continuum X, let X² denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let Ck(X) stand for the hyperspace of members of X² having at most k components. Consider mappings (where B∈Cm(X)) and both defined by A?A∪B. We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for φB). The conditions are related to the contractibility of the corresponding hyperspaces.     

The ω-limit set of a graph map

Let G be a graph and be continuous. Denote by P(f), , ω(f) and Ω(f) the set of periodic points, the closure of the set of periodic points, ω-limit set and non-wandering set of f, respectively. In this paper we show that: (1) v∈ω(f) if and only if v∈P(f) or there exists an open arc L=(v,w) contained in some edge of G such that every open arc U=(v,c)⊂L contains at least 2 points of some trajectory; (2) v∈ω(f) if and only if every open neighborhood of v contains at least r+1 points of some trajectory, where r is the valence of v; (3) ; (4) if , then x has an infinite orbit.