Pre-apartness structures on spaces of functions

Pre-apartness structures are defined on Y^X, where X is an inhabited set and Y a uniform space. These structures clarify the discussion of proximal and uniform convergence in the constructive theory of apartness spaces.     

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

Graphical convergence of continuous functions

Let X and Y be metrizable spaces. We show that convergence of a net of continuous functions 〈f _λ 〉 to a continuous function f in the graph topology for C(X,Y) is equivalent to the uniform convergence of the net of associated distance functionals for the graphs with respect to each compatible metric on X×Y. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement.     

Proximal Set-Open Topologies on Partial Maps

Let X, Y be T ₁ topological spaces. A partial map from X to Y is a continuous function f whose domain is a subspace D of X and whose codomain is Y. Let P(X, Y) be the set of partial maps with domains in a fixed class D. In analogy with the global case, we introduce on P(X, Y), whatever be the nature of the domain class D, new function space topologies, the proximal set-open topologies, briefly PSOTs, deriving from general networks on X and proximity on Y by replacing inclusion with strong inclusion. The PSOTs include the already known generalized compact-open topology on partial maps with closed domains. When domains are supposed closed, the network α closed and hereditarily closed and the proximity δ on Y Efremovic, then the PSOT attached to α and δ is uniformizable iff α is a Urysohn family in X. This revised version was published online in June 2006 with corrections to the Cover Date.     

PHANTOM MAPS,COGROUPS AND THE SUSPENSION MAP

ABSTRACT

The set Ph(X, Y) of pointed homotopy classes of phantom maps from X to Y admits a natural group structure if either Y is a grouplike space or X is a cogroup. In the present paper, the group structure on Ph(X,Y) is examined in the second case. (The first case was examined in an earlier paper.) The results in the two cases are similar—for instance, the group structure turns out to be abelian, divisible and independent of the grouplike structure on Y or the cogroup structure on X—but the techniques used to establish the results differ substantially in the two cases.

In addition, a study of the map g*: Ph(X,Y₁) → Ph(X,Y₂) induced by a map g: Y₁ → Y₂ of grouplike spaces is initiated. A particularly interesting special case of this situation is the suspension map Ph(X, Y) → Ph(X, ΩσY) ? Ph(σX, σY) with Y a grouplike space.     

Priestley duality for order-preserving maps into distributive lattices

The category of bounded distributive lattices with order-preserving maps is shown to be dually equivalent to the category of Priestley spaces with Priestley multirelations. The Priestley dual space of the ideal lattice L of a bounded distributive lattice L is described in terms of the dual space of L. A variant of the Nachbin-Stone-ech compactification is developed for bitopological and ordered spaces. Let X be a poset and Y an ordered space; X ^Y denotes the poset of continuous order-preserving maps from Y to X with the discrete topology. The Priestley dual of L ^P is determined, where P is a poset and L a bounded distributive lattice.     

Measure of Noncompactness of Linear Operators between Spaces of Sequences That Are (N,q) Summable or Bounded

In this paper we investigate linear operators between arbitrary BK spaces X and spaces Y of sequences that are summable or bounded. We give necessary and sufficient conditions for infinite matrices A to map X into Y. Further, the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for A to be a compact operator.     

Ordered Compactifications of Products of Two Totally Ordered Spaces

We describe the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y where X and Y are certain totally ordered topological spaces, and where _o Z denotes the Stone–ech ordered- or Nachbin-compactification of Z. These basic cases are used to illustrate techniques for describing the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y for arbitrary totally ordered topological spaces X and Y. Such products X × Y provide many counterexamples in the theory of ordered compactifications.     

Subadditive Separating Maps

Let X and Y denote compact Hausdorff spaces and let K = R (real numbers) or C(complex numbers). C(X) and C(Y) denote the spaces of K-valued continuous functions on X and Y, respectively. A map H : C(X) C(Y) is separating if fg = 0 implies that HfHg = 0. Results about automatic continuity and the form of additive and linear separating maps have been developed in [1], [2], [3], [4], [5], [7], [8], and [10]. In this article similar results are developed for subadditive separating maps. We show (Theorem 5.11) that certain biseparating, subadditive bijections H are automatically continuous.     

Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:

(i)

Characterize those base spaces X and Y for which all isometries are weighted composition maps.

(ii)

Give a condition independent of base spaces under which all isometries are weighted composition maps.

(iii)

Provide the general form of an isometry, both when it is a weighted composition map and when it is not.

In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.     

Automatic continuity and representation of group homomorphisms defined between groups of continuous functions

Let C(X,G) be the group of continuous functions from a topological space X into a topological group G with pointwise multiplication as the composition law, endowed with the uniform convergence topology. To what extent does the group structure of C(X,G) determine the topology of X? More generally, when does the existence of a group homomorphism H between the groups C(X,G) and C(Y,G) implies that there is a continuous map h of Y into X such that H is canonically represented by h? We prove that, for any topological group G and compact spaces X and Y, every non-vanishing C-isomorphism (defined below) H of C(X,G) into C(Y,G) is automatically continuous and can be canonically represented by a continuous map h of Y into X. Some applications to specific groups and examples are given in the paper.     

Boundedly UC Spaces and Topologies on Function Spaces

A new interesting topology on graphs of partial maps is introduced. This topology can be considered as a natural extension to a non locally compact setting of former topologies defined by P. Brandi, R. Ceppitelli and K. Back, having applications in mathematical economics, differential equations and in the convergence of dynamic programming models. New characterizations of boundedly Atsuji spaces are given by the coincidence of and the topology τ _ucb of uniform convergence on bounded sets on C(X,Y) and by topological properties of .      

The limit space Cc(X,Y) and the connected components of X and Y

Let X and Y be limit spaces (in the sense of FISCHER). For f ? C(X, Y), let [f] denote the subset of C(X, Y), where the maps take the connected components of X into those of Y quite analogously to f. The subspace [f] of the continuous convergence space C_c(X, Y) is written as a product II C_c(X_i, Y_k(i)), where X_i runs through the components of X and Y_k(i) always is the component of Y which contains the set f(X_i). Sufficient conditions for the representation C_c(X, Y) = Σ [f] are given (in terms of the spaces X and Y). Some applications on limit homeomorphism groups are included.     

On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L ^p (μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L ¹(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.     

Remarks on Linear Homeomorphisms of Function Spaces

It is proved that whenever X and Y are completely regular -spaces of pointwise countable type and the spaces C _p(X) and C _p(Y) of real-valued continuous functions on X and Y, respectively, endowed with the topology of pointwise convergence, are linearly homeomorphic, the X is locally compact iff Y is locally compact. This extends the McCoy and Ntantu result.     

On regularly branched maps

Let be a perfect map between finite-dimensional metrizable spaces and p1. It is shown that the space of all bounded maps from X into with the source limitation topology contains a dense G_δ-subset consisting of f-regularly branched maps. Here, a map is f-regularly branched if, for every n1, the dimension of the set is n(dimf+dimY)−(n−1)(p+dimY). This is a parametric version of the Hurewicz theorem on regularly branched maps.     

Research announcement on gaierkirfs method for a class of integra equations of the first kind

Let X and Y be locally convex spaces with K a closed convex cone in X Necessary and sufficient conditions are given for the image AK to be closed in Ywhen A:X→Y is a continuous linear map. This result is used to generalize a theorem of Abrams to infinite dimensional spaces and also to give sufficient conditions for the Hurwicz version of the Farkas lemma for locally convex spaces to hold.     

Densely Continuous Forms in Vietoris Hyperspaces

For countably paracompact normal spaces X and locally compact separable metric spaces Y, a characterization is given for the closure of the set of densely continuous forms from X to Y in the hyperspace of nonempty closed subsets of X × Y under the Vietoris topology. This shows that for such X having no isolated points, every closed subset of X × R that is dense over X can be Vietoris approximated by a semicontinuous function on X.     

Extensions of lipschitz maps into Banach spaces

It is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map fromX intoZ so that wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX. Supported in part by US-Israel Binational Science Foundation and by NSF MCS-7903042. Supported in part by NSF MCS-8102714.